ie352l1_labmanual

Laboratory Exercise 1 Comparative Experiments – Z-test (One-sample mean test) Course Code:

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Instructor:

1. Objective(s): The activity aims to introduce the z-test as another test under the parametric statistics that requires normality of distribution using MiniTab. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 describe the use of z-test in comparing means, sample mean, and population mean, 2.2 solve for the z-value using MiniTab, 2.3 interpret and compare the result in the table of tabular value of the z-test. 3. Discussion: Statistics is all about understanding the role of chance in our measurements and we often want to know what the chances are of obtaining sample means given the population mean is a certain value. The standard error of the mean identifies how much the sample mean varies from sample to sample (it is the standard deviation of the population mean given a particular sample size). The empirical rule tells us that 95% of the time the sample mean will fall within two standard errors of the population mean. We can extend the principle of the empirical rule and use the normal curve to find the probabilities for a given sample mean using a statistical test called the 1-sample z-test. Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be 1

approximated by a normal distribution. Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. For each significance level, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's ttest which has separate critical values for each sample size. Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance known. The Z-test is typically with standardized tests, checking whether the scores from a particular sample are within or outside the standard test performance. The z value indicates the number of standard deviation units of the sample from the population mean. Note that the z-test is not the same as the z-score, although they are closely related. The tabular value of the z test at 0.01 and 0.05 level of significance is shown below: Test

Level of Significance 0.01

0.05

One-tailed

+2.33

+1.645

Two-tailed

+2.575

+1.96

The formula is:

Where: x = sample mean µ = hypothesized value of the population mean σ = population standard deviation n = sample size

4. Resources: MiniTab Software/Manual Textbooks 5. Procedure: Practice Problem: A school principal claimed that the average score of their students in the reading 2

comprehension test should have an average of 75.00, with a standard deviation of 7.5. If 50 randomly selected students have an average of 82.5, use z-test to test the null hypothesis that µ = 75.00 against the alternative hypothesis of µ ≠ 75.00 at 0.05 level of significance. Procedure: 1. Open a blank worksheet in the MiniTab. 2. Choose the Stat option from the menu bar of the Minitab window. 3. Select Basic Statistics > 1 Sample Z test

4. Input the following information given in the problem in the 1-Sample Z (Test and Confidence Interval) dialog box

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5. Click on the Graphs button to select the type of graphical representation needed. Click the OK button to continue.

6. Click on the Options button to define the Confidence Level and the Alternative. Click OK to continue.

7. Click the OK button on the main window to run the analyses. The output will be displayed in the Session window. 6. Data and Results:

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7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

6

Laboratory Exercise No. 2 Hypothesis Testing and Confidence Intervals Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce hypothesis testing and confidence intervals applied to 1-sample t-test. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1

compare the mean data of a sample to a known value using 1-sample T-test

2.2

use a hypothesis test to make inferences about one or more populations when sample data are available.

2.3

quantify the precision of the estimate using confidence interval.

2.4

interpret results and draw conclusions about the output provided by Minitab.

3. Discussion: A hypothesis test uses a sample data to test a hypothesis about the population from which the sample was taken. The 1-sample t-test is one of many procedures available for hypothesis testing in Minitab. For example, to test whether the mean length, measure several rods and the use of mean length of these samples to estimate mean length of the total rod population. Using the information from a sample to make a conclusion about a population is known as statistical inference. Use a 1-sample t-test to determine whether µ (the population mean) is equal to a hypothesized value (the hypothesized mean). The test uses the standard deviation of the sample to estimate σ (the population 7

standard deviation). If the difference between the sample mean and the hypothesized mean is large relatively to the variability of the sample mean, then µ is unlikely to be equal to the hypothesized mean. Use a 1-sample t-test with continuous data from a single random sample. The test assumes the population is normally distributed. However, the test is robust to violations of this assumption, provided the observations are collected randomly and the data are continuous, unimodal, and reasonably symmetric. 4. Resources: MiniTab Software/Manual Training Data Sets Textbooks 5. Procedure: Practice Problem: A cereal manufacturer wants to determine whether the box-filling process is on target. The target fill weight for cereal boxes is 365 grams. Engineers choose six boxes of cereal at random, weigh them, and use the sample data to estimate the mean of the population (the process mean). The manufacturer needs to determine whether the mean weight for the packaging process differs significantly from the target weight of 365 grams. In statistical terms, the process mean is the population mean, or µ (mu). Part 1: 1. Open CEREALBX.MPJ 2. Choose Stat ►Basic Statistics ►1-Sample t

3. Complete the dialog box as shown below. 8

4. Click OK. 5. Interpret the results. 6. Make a decision. 7. Draw conclusions. Part 2: Testing the assumption of normality 1. Choose Stat ►Basic Statistics ►Normality Test

2. Complete the dialog box as shown below. 9

3. Click OK. 4. Interpret the result 5. Draw conclusions. Part 3: Confidence Intervals 1. Choose Stat ►Basic Statistics ►1-Sample t

2. Click Graphs. 3. Complete the dialog box as shown below.

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4. Click OK in each dialog box. 5. Interpret the results 6. Draw conclusions. 6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

12

Laboratory Exercise No. 3 Power and Sample Size Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce the basic information for sample size calculation and power analysis using Minitab. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1

determine the sample size

2.2

evaluate the power to detect the difference of the collected data

3. Discussion: Power is the ability of a test to detect a difference when one exists. A hypothesis test has the following possible outcomes: Null hypothesis Decision Fail to reject Reject

True Correct Decision p=1-ߙ Type I error p=ߙ (power)

False Type II error p=β Correct Decision p=1–β

The power of the test is the probability that you will correctly reject the null hypothesis, given that the null hypothesis is false. Use a power analysis to determine how much power a test has or to design a new test with adequate power. 13

Values To estimate power, you must specify values for any two of the following parameters of the test; Minitab calculates the remaining parameter. 8. Sample sizes – the number of observations in the sample 9. Differences – a meaningful shift away from the target that you are interested in detecting with high probability 10. Power values – the power (probability of rejecting H0 when it is false) that you would like the test to have. 4. Resources: MiniTab Software/Manual Textbooks 5. Procedure: Practice Problem: The engineers are concerned about the results of the fill weight analysis (Laboratory Exercise 2) because of its small sample size. They decide to conduct a power analysis to determine whether they collected enough sample data to detect a difference. They want to be sure the process mean fill weight does not differ from the target weight of 365 grams by more than 2.5 grams. The engineers base the power analysis on the result of t-test from Laboratory exercise 2. 1. Choose File ►New, select Minitab Project, and click OK. 2. Choose Stat ►Power and Sample Size ►1-Sample t. 3. Complete the dialog box as shown below.

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4. Click OK. 5. Interpret the results. 6. Draw conclusions. Part 2: Determining power: With 6 observations, the power of the test was only 0.5377. To have a better chance of detecting a difference, increase the power of the test to at least 0.80 by increasing the sample size. Calculate the sample sizes required to achieve power levels of 0.80, 0.85, 0.90, and 0.95. 6. Choose Stat ►Power and Sample Size ►1-Sample t 7. Complete the dialog box as shown below.

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8. Click OK. 9. Interpret the result 10. Draw conclusions. 6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

17

Laboratory Exercise No. 4 1-Sample t-Test Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce 1-sample t-test used for independent samples as a more powerful test compared with other tests of difference of two independent groups. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 evaluate the difference between a process (population) mean and a target value using 1-sample ttest. 3. Discussion: A one sample t-test measures whether a sample value significantly differs from a hypothesized value. For example, a Movielens researcher might hypothesize it takes 50 seconds for a new user to add a friend to their buddy list. The researcher conducts an experiment and measures how long it takes several new users to perform the task. The one sample t-test measures whether the mean amount of time it took the experimental group to complete the task varies significantly from the hypothesized 50 second value. The one sample t-test requires that the dependent variable follow a normal distribution. When the number of subjects in the experimental group is 30 or more, the central limit theorem shows a normal distribution can be assumed. If the number of subjects is less than 30, the researcher should plot the 18

results and examine whether they appear to follow a normal distribution. If the distribution appears to be non-normal, and/or if the number of test cases is significantly less than 30, then a one sample median test, which does not require a normal distribution, should be used to test the hypothesis. Values to report are the following: the mean of the test group, degrees of freedom for the t-test, t-value, and p value. 4. Resources: MiniTab Software/Manual Training Data Set Textbooks 5. Procedure: Practice Problem: The result of the first power analysis suggest that a larger sample would be useful in evaluating the process. Six observations did not have enough power to detect a 2.5-gram difference. Engineers randomly select 12 boxes of cereal and weigh them. Analyze the new sample to determine whether the process mean is different from 365 grams. 11. Open CEREALBX.MPJ 12. Choose Window ►Worksheet 2. 13. Choose Stat ►Basic Statistics ►1-Sample t. 14. Complete the dialog box as shown below.

15. Click Graphs. 19

16. Check Boxplot of Data. 17. Click OK in each dialog box 18. Interpret the results 19. Draw conclusions. Part 2: The 1-Sample t-test assumes the data are sampled from a normally distributed population. Use a normality test to determine whether the assumption of normality is valid. 7. Choose Stat ►Basic Statistics ►Normality Test 8. Complete the dialog box as shown below.

9. Click OK. 10. Interpret the result 11. Draw conclusions. 6. Data and Results:

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7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills. Members follow safety precautions at all times.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment. Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members finish ahead of time with complete data and time to revise data. Members have defined Members are on tasks responsibilities most of and have the time. Group responsibilities at all conflicts are times. Group conflicts cooperatively managed are cooperatively most of the time. managed at all times. Clean and orderly Clean and orderly workplace with workplace at all times occasional mess during during and after the and after the experiment. experiment. Members require Members do not need occasional supervision to be supervised by the by the teacher. teacher. TOTAL SCORE RATING= x 100%

22

Laboratory Exercise No. 5 Power and Sample Size for 2-Sample t-Test Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce basic ideas of power and sample size calculations for 2-sample t-Test. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1

test for a difference between two population means using a 2-sample t-tes

2.2

determine the sample size required to detect an effect of a given size with a given degree of confidence.

3. Discussion: In a 2-sample t-test, power is the probability that you will detect a difference between the two means when they actually differ while the sample size is the number of samples per group that you need to achieve a specified power. The analysis can be used either: before collecting the data, to determine the sample size or after collecting the data, to evaluate the power to detect a difference between means. Power and sample size can determine the following: 12. The sample size per group that you need to detect a difference between means with a specified power 13. The power of a test to detect a difference between means based on a specified sample size 14. The size of a detectable difference with a specified power and sample size Determining the sample size for 2-sample t-test: 23

Sample sizes – do not enter sample size when you want to determine the sample size. Values of difference and standard deviation – the power of a test depends on the difference you want to detect relative to the standard deviation. To detect a 1-standard deviation (or 1-sigma) difference, enter a difference of 1 and -1, and a standard deviation of 1. Power values – enter the desired power value(s). Power values higher than 0.80 are typically considered acceptable. 4. Resources: MiniTab Software/Manual Textbooks 5. Procedure: Practice Problem: A calculator manufacturer is selecting a plastic supplier. The quality team has a policy for critical quality metrics that states: “Assuming similar variability and costs, mean strengths more than one standard deviation apart are an important difference.” Determine the sample size needed to detect a difference of one standard deviation between two suppliers with similar variability. (Minitab assumes equal variability in the sample size calculation.) The power to detect this difference should be at least 80%. 20. Choose File ►New, select Minitab Project , and click OK 21. Choose Stat ►Power and Sample Size ►2-Sample t. 22. Complete the dialog box as shown below.

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23. Click OK 24. Interpret the results 25. Draw conclusions. 6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Messy workplace during and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Neatness and Orderliness

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

26

Laboratory Exercise No. 6 2-Sample t-Test Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce basic ideas of power and sample size calculations for 2-sample t-Test. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1

test for a difference between two population means using a 2-sample t-test

2.2

determine the sample size required to detect an effect of a given size with a given degree of confidence.

3. Discussion: An independent 2-sample t-test helps determine whether two population means are different. The test uses the sample standard deviations to estimate σ for each population. If the difference between the sample means is large relative to the estimated variability of the sample means, then the population means are unlikely to be the same. Independent 2-sample t-test can also be used to evaluate whether the means of two populations are different by a specific amount. When to use an independent 2-sample t-test? Use an independent 2-sample t-test with continuous data from two independent random samples. Samples are independent if observations from one sample are not related to the observations from the other sample. The test also assumes that the data come from normally distributed populations. 27

However, the test is robust to violations of this assumption, provided the observations are collected randomly and the data are continuous, unimodel, and reasonably symmetric. Why use an independent 2-sample t-test? An independent 2-sample t-test answers questions such as: 3 Are the means of a product characteristic between two suppliers comparable? 4 Is one formulation of a product better on average than other? 4. Resources: MiniTab Software/Manual Training Data Set, Textbooks 5. Procedure: Practice Problem: A calculator manufacturer is selecting a plastic supplier. Using a sample size of 20 plastic pellets from each supplier, the manufacturer must compare samples from the two suppliers for strength. 26. Open PLASTIC.MPJ 27. Choose Stat ►Basic Statistics ►2-Sample t. 28. Complete the dialog box as shown below.

29. Click Graphs 30. Check Individual value plot and Boxplots of data 28

31. Click OK in each dialog box 32. Interpret the results 33. Draw conclusions. Part 2: Testing the normality assumption: The 2-sample t-test assumes the data are sampled from normally distributed populations. 2

Choose Stat ►Basic Statistics ►Normality Test

3

In Variable, enter SupplrA

4

Click OK.

5

Choose Stat ►Basic Statistics ►Normality Test

6

In Variable, enter SupplrB

7

Click OK.

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Interpret the results

9

Draw conclusions.

Part 3: Comparing variances: The 2-sample t-test compares the means of two populations. Often it is of interest to know whether the variances (or standard deviations) of two groups are different. 1. Choose Stat ►Basic Statistics ►2 Variances 2. Complete the dialog box as shown below.

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3. Click OK. 4. Interpret the results. 5. Draw conclusions. 6. 6. Data and Results:

7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills. Members follow safety precautions at all times.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment. Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members finish ahead of time with complete data and time to revise data. Members have defined Members are on tasks responsibilities most of and have the time. Group responsibilities at all conflicts are times. Group conflicts cooperatively managed are cooperatively most of the time. managed at all times. Clean and orderly Clean and orderly workplace with workplace at all times occasional mess during during and after the and after the experiment. experiment. Members require Members do not need occasional supervision to be supervised by the by the teacher. teacher. TOTAL SCORE RATING= x 100%

31

Laboratory Exercise No. 7 Paired t-Test Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce basic ideas of power and sample size calculations for 2-sample t-Test. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1

test for a difference between two population means using a 2-sample t-test

2.2

determine the sample size required to detect an effect of a given size with a given degree of confidence.

3. Discussion: A paired t-Test helps determine whether the mean differences between paired observations is significant. Statistically, the paired t-test is equivalent to performing a 1-sample t-test on the differences. A paired ttest also helps you to evaluate whether the mean difference is equal to a specific value. Paired observations are related. Examples include: 1. Weights recorded for individuals before and after an exercise program 2. Measurements of the same part taken with two different measuring devices. Paired t-test with a random sample of paired observations. The test also assumes that the paired differences come from a normally distributed population. However, the test is robust to violations of this 32

assumption, provided the observations are collected randomly and the data are continuous, unimodal, and reasonably symmetric. Why use a paired t-test? A paired t-test answers questions such as: 1. Does a new treatment result in a difference in the product? 2. Do two different instruments provide similar measurements for the same sample? 4. Resources: MiniTab Software/Manual Training Data Set Textbooks 5. Procedure: Practice Problem: A consumer group wants to determine whether drivers can park one car more quickly than the other. Because the data are paired (each individual parked both cars), use a paired t-test to teatv the following hypothesis: H0: The mean difference between paired observations in the population is zero. H1: The mean difference between paired observations in the population is not zero. Use the default confidence level of 95%. Display individual value plots and boxplots to help visualize the data. 1. Open CARCTL.MPJ 2. Choose Stat ►Basic Statistics ►Paired t. 3. Complete the dialog box as shown below.

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4. Click Graphs 5. Check Individual value plot and Boxplots of differences. 6. Click OK in each dialog box 7. Interpret the results 8. Draw conclusions.

Part 2: Testing the normality: The paired t-test 1. Choose Stat ►Basic Statistics ►Normality Test 2. In Variable, enter SupplrA 3. Click OK. 4. Choose Stat ►Basic Statistics ►Normality Test 5. In Variable, enter SupplrB 7. Click OK. 8. Interpret the results 9. Draw conclusions. Part 3: Checking for Normality: the paired t-test is actually a 1-sample t-test on the pair wise difference. Therefore, the pair wise differences must satisfy the 1-sample t-test assumptions, including normality. 34

Before checking for normality, store the pair wise differences in the worksheet. 1. Choose Stat ►Basic Statistics ►2 Variances 2. Complete the dialog box as shown below.

3. Click OK. 4. Interpret the results. 5. Draw conclusions. 6. Data and Results:

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7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

37

Laboratory Exercise No.8 Correlation Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 Evaluate the linear relationship between two variables using scatterplot, correlation, and fitted line plot. 2.2 Analyze and interpret results and draw conclusions about the output provided by Minitab. 3. Discussion: The sample correlation coefficient , r, measures the degree of linear association between two variables (the degree to which one variable changes with another). A positive correlation indicates that both variables tend to increase or decrease together. A negative correlation indicates that, as one variable increases, the other tends to decrease. Use correlation when you have data for two continuous variables and wish to determine whether a linear relationship exists between them. The correlation does not tell you whether the variables are related in a non linear fashion. Some statisticians argue that correlation should not be used if one variable is a dependent response of the other. Correlation can help answer questions such as 38

1. Are two variables related in a linear manner? 2. What is the strength of the relationship? Example A. Is there a linear relationship between dollars spent on training and customer satisfaction ratings? B. What is the relationship between revenue and the number of sales calls made? Additional Considerations Correlation quantifies the degree of linear association between two variables. A strong correlation does not imply a cause-and-effect relationship. For example, a strong correlation between two variables may be due to the influence of a third variable not under consideration. A correlation coefficient close to zero does not necessarily mean no association. The variables may have a nonlinear association. Always plot the data so that you can identify nonlinear relationships when they are present. Some statisticians argue that correlation should not be used if one variable is a dependent response of the other. Correlation assumes that the values of both variables are free to vary. Correlation is not appropriate if you fix the values of one variable to study changes in another. 4. Resources: MiniTab Software/Manual Training Data Sets Textbooks 5. Procedure: Practice Problem: The sales department for a software company wants to determine whether a relationship exists between the number of sales calls made and the revenue earned. Analysts record the number of sales calls and the revenue earned each day for a period of 420 days. Variable

Description 39

Revenue

Daily Revenue in thousands of dollars, rounded to the nearest dollar

Sales Calls Number of sales calls made each day. Part 1: 1. Open SoftRev1.MPJ 2. Choose Graph ►Scatterplot 3. Choose Simple, then click OK 4. Complete the dialog box as shown below.

5. Click OK. 6. Interpret the results Part 2: Calculating the correlation 11. Choose Stat ►Basic Statistics ►Correlation

40

12. Complete the dialog box as shown below.

13. Click OK 14. Interpret the results 15. Draw conclusions. 6. Data and Results:

41

7. Data Analysis and Conclusion:

42

8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Messy workplace during and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Neatness and Orderliness

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

43

Laboratory Exercise No.9 Simple Linear Regression Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to measure the degree of linear association between two variables using graphs and correlation Model the relationship between a continuous response variable and one or more predictor variables. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 Evaluate the linear relationship between two variables using scatterplot, correlation, and fitted line plot. 2.2 Analyze and interpret results and draw conclusions about the output provided by Minitab. 3. Discussion: Simple Linear Regression examines the relationship between a continuos response variable (y) and one predictor variable (x) . The general equation for a simple linear regression model is:

Y   O  1   Where Y is the response, X is the predictor,  O is the intercept (the value of Y when X equals zero), 1 is the slope and  is random error. Use simple linear regression when you have a continuos y and one predictor , x. The following conditions 44

should also be met: 1. X can be ordinal or continuos 2. In theory, x should be fixed by the investigator. In practice, however, it is often allowed to vary. 3. Any random variation in the measurement of x is assumed to be negligible compared to the range in which x is measured. The y-values obtained in your sample differ from those predicted by the regression model (unless all points happen to fall on a perfectly straight ine). These differences are called residuals. To confirm that the analysis is valid, verify all assumptions about the model error term. Use residual plots to check that the errors have the following characteristics: 1. Normally distributed 2. Constant variance for all fitted values 3. Random over time Simple Linear Regression can help answer the following questions such as 1. How important is x in predicting y? 2. What value can you expect for y when x is 5? 3. How much does y change if x increases by one unit? For example, Is the number of mistakes made in processing loans related to cycle time? What salary can you expect to make with five years experience in a particular field? How much does salary increase for every additional year of experience? S is an estimate of the average variability about the regression line. S is the positive square root of the mean square error (MSE). For a given problem, the better the equation predicts the response, the lower S is. 2

R (R  Sq ) R 2 is the proportion of variability in the response that is explained by the equation. Acceptable values for R

2

vary depending on the study. For example for engineers studying chemical reactions may require an

R 2 of 90% or more. However, someone studying human behavior ( which is more variable) may be satisfied with much lower R 2 values. 45

2

R adjusted (R  q (adj)) S 2

R adjusted is sensitive to the number of terms in the model and is important when comparing models with different number of terms. The Least Squares regression line The coefficients for the regression equation are chosen to minimize the sum of the squared differences between the response values observed in the sample and those predicted by the equation. In other words the squared vertical distances between the points and line are minimized. The result is called the Least squares regression line. Confidence and prediction bands Confidence bands provide the estimated range in which the mean response for a given value of the predictor is expected to fall. Prediction bands provide the estimated range in which a single new observation for a given value of the predictor is expected to fall. Analysts want to be confident that the mean and the individual points of the y-variable, Revenue, fall within certain limits of variability. Use the default confidence level of 95% Confidence Interval The 95% confidence interval defines a likely range of values for the population mean of y. For any given value of x, you can be % confident that the population mean for y is between the indicated lines. Prediction interval The 95% prediction interval defines a likely range of y values for future individual observations. For any given value of x, you can be 95% confident that the corresponding value of y for a single future observation is between the indicated lines. Note : The prediction interval is always wider than the confidence interval because of the added uncertainty 46

involved in predicting a single response versus the mean response. Residuals The residuals for each observation is the difference between the observed value of the response and the value predicted by the model ( the fitted value). For example, if the observed response value is 12 and the model predicts 10, the residual is 2. Assumptions 1. To confirm that the analysis is valid. Verify all assumptions about the model error term. Use residual plots to check that the errors have the following characteristics. 2. Normally distributed 3. Constant variance for all fitted values 4. Random over time Normal Probability Plot The normal probability plot should roughly follow a straight line. Use this plot to verify that the residuals do not deviate substantially from a normal distribution. Histogram Use the normal probability plot to make decisions about the normality of the residuals. With a reasonably large sample size, The histogram displays compatible information with the normal probability plot The histogram of the residuals should appear approximately bell-shaped with no unusual values or outliers. Use the histogram as an exploratory tool to learn about the following characteristics of the data. -Typical values, spread or variation, and shape -Unusual values in the data Residual versus fits Use the plot of the residuals versus fits to verify that the residuals are scattered randomly about zero. This pattern…….

Indicates

………………….. 47

Curvilinear

A quadratic term may be missing from the model

Fanning or uneven spread Of residuals across the different fitted values

Non constant variance of the residuals

Points far away from zero relative to other Data points

Outliers exist

Residual versus order The plot of the residuals versus order displays the residuals in the order of data collection (provided the data were entered in the same order in which they were collected.) If the data collection order affects the results, residuals near each other may be correlated , and thus , not independent. This pattern……. Residuals are not randomly scattered around zero Residuals are randomly scattered around zer Points far away from zero

Indicates

…………………..

Residuals are not independent over time Residuals are independet Outliers exist

Additional Considerations 1. Be careful when using regression analysis to assert that changes in the predictor values were fixed at predetermined levels in a controlled experiment. If the values of the predictors are allowed to vary randomly, other factors may influence both the predictors and the response. 2. Do not apply regression results to values of x that are outside the sample range. The relationship between Sales calls and Revenue may be very different for sales calls above 168. 3. Be alert for outliers when using regression procedures. Some outliers (called high leverage points) have a large effect on the calculation of the least squares regression line. In such cases, the line may no longer represent the rest of the data very well. 4. Time order trends in the data can violate the assumption of independence,. A run chart or individual chart is a useful tool for detecting such efforts. 4. Resources: 48

MiniTab Software/Manual Training Data Sets Textbooks 5. Procedure: Practice Problem: The sales department for a software company wants to determine whether a relationship exists between the number of sales calls made and the revenue earned. Analysts record the number of sales calls and the revenue earned each day for a period of 420 days.Determine the effect of Sales calls on Revenue. Use fitted line plot to calculate and plot the regression equation. Variable

Description

Revenue

Daily Revenue in thousands of dollars, rounded to the nearest dollar

Sales Calls Number of sales calls made each day. Part 1: Fitted Line Plot 1. Open SoftRev1.MPJ 2. Choose Stat ►Regression ►Fitted Line Plot 3. Complete the dialog box as shown below.

4. Click OK. 49

5. Interpret the results. 6. Evaluate the results using the ANOVA results to evaluate whether the simple regression model is useful for predicting revenue. State Hypothesis 7. Interpret the p-value (P) . 8. Make a conclusion. Part 2: Adding confidence and prediction bands 1. Choose Stat ►Regression ►Fitted Line Plot or Press (Ctrl)+(E) 2. Click Options 3. Complete the dialog box as shown below.

4. Click OK 5. Click Graphs 6. Complete the dialog box shown below

50

7. Click OK in each dialog box. 8. Interpret Results 5. Normal Probability Plot 6. Histogram 7. Residual versus fits 8. Residual versus order 9. Make conclusions 6. Data and Results:

51

7. Data Analysis and Conclusion:

52

8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Messy workplace during and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Neatness and Orderliness

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

53

Laboratory Exercise No. 10 Multiple Linear Regression Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to measure the degree of linear association between two variables using graphs and correlation Model the relationship between a continuous response variable and one or more predictor variables. 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 Evaluate the linear relationship between two variables using scatterplot, correlation, and fitted line plot. 2.2 Analyze and interpret results and draw conclusions about the output provided by Minitab. 3. Discussion: Multiple Linear regression examines the relationship between a continuous response variable (Y) and more than one predictor variable (X) . The general equation for a multiple regression model is: Y   0  1 X 1   2 X 2   3 X 3  .......   Where y is the response,  0 is the intercept, each xi is a predictor variable with a slope of  i and  is random error. Use multiple linear regression when you have a continuous y and more than one x. 54

1. X can be categorical , ordinal, or continuos. 2. Any random variation in the measurement of x is assured to be neglible compared to the range within which x is measured. Before accepting the results of a regression analysis, verify that the following assumptions about the errors are valid: 1. They must be independent 2. They must be normally distributed 3. They must have a constant variance across all values of x. 4. They are not correlated with a predictor. Multiple Linear regression can help answer the following question such as: 1. How important are the x – variables in predicting y? 2. What value is expected for y when x1 is 20 and x2 is 3? 3. How much will y change if X3 increases by one unit (when x1 and x2 are fixed)? For example, 1. How do flight- delay length and the number of empty seats relate to customer satisfaction rating? 2. How is the satisfaction affected by a flight delays and lost luggage? 4. Resources: MiniTab Software/Manual Training Data Sets Textbooks 5. Procedure: Practice Problem: You are selling your house, and want to establish a fair sale price. Data Collection The following data were collected for a random sample of houses sold in 1991: 1. Sale Price 2. Size of the House 3. Number of Bedrooms 4. Age of the house 55

5. Area in which the house was built 6. Real estate agent Variabl e Price Bedroo ms

Size Age Are a Agen cy

Description Sale price of the house in thousands of dollars Number of bedrooms in the house

Size of the house in square feet Age of the house in years Area in which the house was built (Dallas, Fort Worth, or Suburbs) Selling agent (ClientFirst or Other)

Part 1:Using of matrix plot to examine potential relationships between sales price, size of house, number of bedrooms, and age of house. 1. Open HouseSale.MPJ 2. Choose Graph ►Matrix Plot

3. Under Matrix of Plots, choose Simple, then click OK

56

4. In Graph Variables, enter Price, Bedrooms, Size Age

5. Click Matrix Options. 6. Under Matrix Options, choose Lower left.

57

7. Click OK in each dialog box 8. Interpret the results

Part 2: Calculate the correlation coefficient for each pair of variables. 1. Choose Stat ►Basic Statistics►Correlation

58

2.

In Variables, enter Price, Bedrooms, Size and Age, then click OK

3. Interpret the results. Part 3: Use General Regression to identify an appropriate model for the data 59

1. Choose Stat ►Regression ►General Regression

2. In Response, enter Price 3. In Model, enter Bedrooms, Size , Age, Area, Agency. 4. In Categorical predictors, enter Area , Agency. 5. Click OK

6. Interpret the results. 60

Part 4: Refit the model excluding the variable Age. 1. Choose Stat ►Regression ►General Regression 2. In Model, remove Age 3. Click OK 4. Interpret the results Part 5: Refit the model excluding the variable Agency. 1. Choose Stat ►Regression ►General Regression 2. In Model, remove Agency 3. In Categorical Predictors, remove Agency 4. Click OK 5. Interpret the results Part 6: Refit the model excluding the variable Bedroom 1. Choose Stat ►Regression ►General Regression 2. In Model, remove Bedrooms 3. Click Graphs 4. Under Residual Plots, choose Four in one 5. Click OK in each dialog box. 6. Interpret the results 6. Data and Results:

61

7. Data Analysis and Conclusion:

62

8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Messy workplace during and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Neatness and Orderliness

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

63

Laboratory Exercise No. 11 One way Analysis of Variance Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

` 1. Objective(s): The activity aims to introduce one way analysis of variance by comparing means of samples collected at different levels using a one-way model and Interpret the main effects plot and multiple comparisons 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 Evaluate differences between group means for a single factor using one-way ANOVA 2.2 Interpret results and draw conclusions about the output provided by Minitab. 3. Discussion: Analysis of variance (ANOVA) Tests the hypothesis that the means of two or more populations are equal. ANOVAs evaluate the importance of one or more factors by comparing the response variable means at the different factor levels. The null hypothesis states that all population means (factor level means) are equal while the alternative hypothesis states that at least one is different. To run an ANOVA, you must have a continuous response variable and at least one categorical factor with two or more levels. ANOVAs require data from normally distributed populations with roughly equal variances between factor levels.

64

For example, you design an experiment to assess the durability of four experimental carpet products. You place a sample of each carpet type in ten homes and you measure durability after 60 days. Because you are examining one factor (carpet type) you use a one-way ANOVA. If the p-value is less than your alpha, then you conclude that at least one durability mean is different. To further explore the differences between specific means, use a multiple comparison method such as Tukey's. The name "analysis of variance" is based on the manner in which the procedure uses variances to determine whether the means are different. The procedure works by comparing the variance between group means versus the variance within groups as a method of determining whether the groups are all part of one larger population or separate populations with different characteristics. Minitab has different types of ANOVAs to allow for additional factors, types of factors, and different designs to suit your specific needs. ANOVA type One-way

Model and Design Properties One fixed factor (levels set by investigator) which can have either an unequal (unbalanced) or equal (balanced) number of observations per treatment combination.

Two-way

Two fixed factors and requires a balanced design.

Balanced

Model may contain any number of fixed and random factors (levels are randomly selected), and crossed and nested factors, but requires a balanced design.

General

Expands on Balanced ANOVAs by allowing unbalanced designs and covariates

Linear Model

(continuous variables).

One way Anova The one way ANOVA (analysis of variance) procedure is a generalization of the independent samples of T- test. Unlike the T-test. However, You can use one way ANOVA to analyze the means of more than two 65

groups (samples)at once. Use one way ANOVA ( also called single-factor ANOVA) when you have continuous response data for two or more fixed levels of single factor. Before accepting the results of an ANOVA, you must verify that the following assumptions about the errors are valid for your data. They must be: 1. Be independent (and thus random) 2. Not deviate substantially from a normal distribution 3. Have constant variance across all factor levels One way ANOVA can help answer questions such as: 1. Are all branches of your company achieving comparable customer satisfaction ratings? 2. Do treatment group means differ? For example: 1. Do mean customer satisfaction ratings differ between a company’s branches in New Hamphshire, Maine, and Vermont? 2. Which of the three training courses is the most successful in decreasing mean application processing errors? Dot plot A dot plot gives a first look at the data to graphically compare the central tendencies and spreads for the 3 commission types. This graph can also reveal whether outlying data points are present and need to be investigated. Degrees of Freedom The degrees of freedom (DF) Statistic measures how much “independent” information is available to calculate each sum of squares (SS): 1. DF factor  k  1, where k is the number of factor levels 2. DFerror  n  k , where n is the total number of observations 3. DFTotal  n  1, 66

Sum of Squares The sum of squares (SS) measures the amount of variability each source contributes to the data. Notice that, SS Total  SS between  SS error Mean Square The mean square (MS) for each source is equal to the SS divided by the DF. F statistic F is the ratio of the variability contributed by the factor to the variability contributed by error.  MS factor   F   MS error  1. If between- group variability is similar to within group variability , F is close to 1, indicating that the factor does not affect the responsible variable 2. If between group variability is larger than within group variability, F is greater than 1. P value A large F suggests that the factor level means are more different than expected by chance, thus the Pvalue is small. Individual Confidence Interval When the p-value in the analysis of variance table indicates a difference among the factor level means, the table individual confidence intervals is sometimes used to assess the differences. 4. Resources: MiniTab Software/Manual Training Data Sets Textbooks 5. Procedure: Practice Problem: Sales representatives at a software company are offered one of three types of salaries: commission, fixed, and a combination of fixed and commission (mixed). The manager of the sales department wants to compare the revenue earned for different salary types. 67

Data Collection The manager records the salary type and revenue earned by each sales representative in a four-month period. Variable Revenue Salary Type

Description Revenue earned in dollars by each sales representative Type of salary received by each sales representative (Commission , Fixed , Mixed)

Part 1: Compare Distributions using Dotplot 1. Open Commission.MPJ 2. Choose Graph ►Dotplot

3. Under One Y, Choose With Groups, then click OK.

4. Complete the dialog box shown below 68

5. Click OK 6. Interpret the results. Part 2 : Perform the one-way ANOVA 1. Choose Stat ►ANOVA ►One-Way 2. Complete the dialog box as shown below.

3. Click Graphs. 4. Under Residual Plots, choose Four in one. 5. Click OK in each dialog box 6. Interpret the results. Ensure that the results are valid, determine whether all the assumptions about the residuals have been met. 69

6. Data and Results:

7. Data Analysis and Conclusion:

70

8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

71

Laboratory Exercise No. 12 Analysis of Variance ( General Linear Model using Tukey-Kramer Method) Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce one way analysis of variance by comparing means of samples collected at different levels using a one-way model and Interpret the main effects plot and multiple comparisons 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 Evaluate differences between group means for a single factor using one-way ANOVA and General Linear Model 2.2 Interpret results and draw conclusions about the output provided by Minitab. 3. Discussion: Tukey's method Used in ANOVA to create confidence intervals for all pairwise differences between factor level means while controlling the family error rate to a level you specify. It is important to consider the family error rate when making multiple comparisons because your chances of making a type I error for a series of comparisons is greater than the error rate for any one comparison alone. To counter this higher error rate, Tukey's method adjusts the confidence level for each individual interval so that the resulting simultaneous confidence level is equal to the value you specify. For example, you are measuring the response times for memory chips. You sampled 25 chips from five different manufacturers. The ANOVA resulted in a p-value of 0.01, leading you to conclude that at least one of the manufacturer means is different from the others. 72

You decide to look at all 10 comparisons between the five plants to determine specifically which means are different. Using Tukey's method, you specify that the entire set of comparisons should have a family error rate of 0.05 (equivalent to a 95% joint confidence level). Minitab calculates that the 10 individual confidence levels need to be 99.35% in order to obtain the 95% joint confidence level. These wider Tukey confidence intervals provide less precise estimates of the population parameter but limit the probability that one or more of the confidence intervals does not contain the true difference to a maximum of 5%. Understanding this context, you can then look at the confidence intervals to see if any do not include zero, suggesting a significant difference. Confidence intervals with 95% individual confidence levels

Confidence intervals with 99.35% individual confidence levels to obtain a 95% joint confidence level using Tukey's

Comparison of 95% confidence intervals (left) to the wider 99.35% confidence intervals used by Tukey's in the above example (right). The reference line at 0 illustrates how the wider Tukey confidence intervals can change your conclusions. onfidence intervals that contain zero suggest no difference. (Only 5 of the 10 comparisons are sh own due to space considerations.) Additional Considerations 1. Comparing multiple factor levels with a single ANOVA is preferable to comparing two levels at a time with separate two-sample t-tests. Extra tests would increase the chances of Type I error (rejecting Ho when Ho is actually true.) 2. The assumption of independence for ANOVA is critical. If observations are symmetrically affected by factors other than the one you are studying (including tinme order effects), the results of one way ANOVA may be meaningless. 3. The assumption of normality for ANOVA is generally not crucial, especially if the sample sizes are large. 4. Resources: MiniTab Software/Manual Training Data Sets Textbooks 73

5. Procedure: Practice Problem: Sales representatives at a software company are offered one of three types of salaries: commission, fixed, and a combination of fixed and commission (mixed). The manager of the sales department wants to compare the revenue earned for different salary types. Data Collection The manager records the salary type and revenue earned by each sales representative in a four-month period.

Variabl e Revenue Salary Type

Description Revenue earned in dollars by each sales representative Type of salary received by each sales representative (Commission , Fixed , Mixed)

Part 1: Understanding the Effects 1. Open Commission.MPJ 2. Choose Stat ►ANOVA ►General Linear Model

3. In Responses, enter Revenue 4. Click Factor Plots

74

5. Complete the dialog box as shown below

6. Click OK 7. Click Comparisons 8. Complete the dialog box as shown below.

75

9. Click OK.in each dialog box 10. Interpret the results. 11. Make a decision. 12. Draw conclusions. 6. Data and Results:

76

7. Data Analysis and Conclusion:

77

8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Neatness and Messy workplace during Orderliness and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

78

Laboratory Exercise No. 13 Analysis of Variance ( General Linear Model application to conduct a one-way ANOVA) Course Code:

Program:

Course Title:

Date Performed:

Section:

Date Submitted:

Members:

Instructor:

1. Objective(s): The activity aims to introduce one way analysis of variance by comparing means of samples collected at different levels using a one-way model and Interpret the main effects plot and multiple comparisons 2. Intended Learning Outcomes (ILOs): The students shall be able to: 2.1 Evaluate differences between group means for a single factor using General Linear Model to conduct a one-way ANOVA 2.2 Interpret results and draw conclusions about the output provided by Minitab. 3. Discussion: . Use General Linear Model (GLM) to perform univariate analysis of variance with balanced and unbalanced designs, analysis of covariance, and regression, for each response variable. Calculations are done using a regression approach. A "full rank " design matrix is formed from the factors and covariates and each response variable is regressed on the columns of the design matrix. You must specify a hierarchical model. In a hierarchical model, if an interaction term is included, all lower order interactions and main effects that comprise the interaction term must appear in the model. Factors may be crossed or nested, fixed or random Covariates may be crossed with each other or with factors, or nested within factors. You can analyze up to 50 response variables with up to 31 factors and 50 79

covariates at one time Balanced ANOVA and general linear model (GLM) are ANOVA procedures for analyzing data collected with many different experimental designs. Your choice between these procedures depends upon the experimental design and the available options. The experimental design refers to the selection of units or subjects to measure, the assignment of treatments to these units or subjects, and the sequence of measurements taken on the units or subjects. Both procedures can fit univariate models to balanced data with up to 31 factors. Here are some of the other options:

Can fit unbalanced data Can specify factors as random and obtain expected means squares Fits covariates Performs multiple comparisons Fits restricted/unrestricted forms of mixed model

Balanced GLM ANOVA no yes yes yes no no yes

yes yes unrestricted only

You can use balanced ANOVA to analyze data from balanced designs Your design must be balanced to use balanced ANOVA, with the exception of a one-way design. A balanced design is one with equal numbers of observations at each combination of your treatment levels. A quick test to see whether or not you have a balanced design is to use Stat > Tables > Cross Tabulation and Chi-Square. Enter your classification variables and see if you have equal numbers of observations in each cell, indicating balanced data. 4. Resources: MiniTab Software/Manual Training Data Sets Textbooks 5. Procedure: Practice Problem: The manager of a call center for a software firm wants to know whether the center needs the same number of people answering the phones each day of the week. 80

Data Collection The number of customer calls to the technical support department is recorded for 205 business days (MonFri) Varia ble Date Week Calls

Description Business date on which data were recorded Day of the week (Mon-Fri) Number of calls to technical support

Part 1: Creating Dotplot to show the distribution for the five days. 1. Open SupCalls.MPJ 2. Choose Graph►Dotplot ►Under One Y, choose With groups, then click OK.

3. In Graph variables, enter Calls 4. In Categorical variables for grouping, enter Weekday 5. Click OK.

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6. Interpret the results 7. Draw Conclusions. Part 2: Fit a general linear model to the data 1. Choose Stat ►ANOVA ►General Linear Model 2. In Response, enter Calls. In Model, enter Weekday

3. Click Graphs, then choose Four in One

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4. Click OK in each dialog box 5. Interpret the results Part 3: Create time series plots for each business day 4. Choose Graph ►Time Series Plot 5. Choose Simple , then click OK

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6. In Series, enter Calls 7. Click Multiple Graphs, then choose the By Variables tab. 8. In By Variables with groups in separate panels, enter Weekday.

9. Click OK in each dialog box 10. Interpret the results Part 4. Create main effects plot for the days of the week and conduct Tukey’s pairwise comparison to determine which weekdays have significantly different means from each other. 6. Choose Stat ►ANOVA ►General Linear Model

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7. Click Factor Plots. 8. Under Main Effects Plot, enter Weekday, then click OK.

9. Click Comparisons. 10. In Terms, enter Weekday, Check Test

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11. Click OK in each dialog box 12. Interpret the results (Examine Tukey comparisons.) 13. Draw Conclusions Part 5: Conduct a test for equal variances to determine if week to week variability is different for different weekdays. 3. Choose Stat ►ANOVA ►Test for Equal Variances

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4. In Response, enter Calls. 5.

In Factors, enter Weekday

6. Click OK in each dialog box 7. Interpret the results 6. Data and Results:

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7. Data Analysis and Conclusion:

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8. Assessment (Rubric for Laboratory Performance): TIP-VPAA–054D Revision Status/Date:0/2009 September 09

CRITERIA

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES RUBRIC FOR LABORATORY PERFORMANCE BEGINNER ACCEPTABLE PROFICIENT 1 2 3

Laboratory Skills Manipulative Members do not Skills demonstrate needed skills. Experimental Members are unable to Set-up set-up the materials.

Members occasionally demonstrate needed skills. Members are able to set-up the materials with supervision. Members occasionally demonstrate targeted process skills.

Members always demonstrate needed skills. Members are able to set-up the material with minimum supervision. Members always demonstrate targeted process skills.

Process Skills

Members do not demonstrate targeted process skills.

Safety Precautions

Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher. Messy workplace during and after the experiment.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time. Clean and orderly workplace with occasional mess during and after the experiment. Members require occasional supervision by the teacher.

Members finish ahead of time with complete data and time to revise data. Members are on tasks and have responsibilities at all times. Group conflicts are cooperatively managed at all times. Clean and orderly workplace at all times during and after the experiment.

Work Habits Time Management/ Conduct of Experiment Cooperative and Teamwork

Neatness and Orderliness

Ability to do Members require independent supervision by the work teacher. Other Comments/Observations:

SCORE

Members do not need to be supervised by the teacher. TOTAL SCORE RATING= x 100%

89