ECOR 1010 – INTRODUCTION TO ENGINEERING
XUELI ZHANG
ASSIGNMENT # 5
Assignment Title: Univariate Data TO Marking TA: Name: Nicholas Bourgeois Email:
[email protected]
Lab Section: C5 Room: ME2556 – Carleton University
FROM: Email:
[email protected] Student Number: 100878403
Number of Figures and Tables (Including handwritten ones): [Total Number of Figures and Tables] Last Date and Time of Revision: 07 November 2012
Zhang, Xueli
INTRODUCTION The purpose of this lab is to determine the probability that the production process will produce knobs with acceptable dimensions (0.137" to 0.143"). The results of sampling are recorded in the accompanying Production Record Sheet and statistics are reported in the appendices.
MATERIALS AND METHODS The production data of knob size was recorded in a Production Record Sheet. An Excel spread sheet was used to analyze the data, and the sample mean, sample variance and standard deviation, standard error, and the 90% confidence interval for the population mean were calculated. The same statistics were also calculated by hand. A frequency histogram was used to visually analyze the data. Two estimates for the fraction of knobs that will be within the acceptable range were calculated. First, measurements in the sample were used to estimate the proportion of acceptable knobs produced in the acceptable range, with a 95% confidence interval. Second, the distribution of measurements was assumed to be a normal Gaussian distribution and the z-tables were used to estimate the proportion of acceptable knobs, with a 99% confidence interval. Using Excel’s Random Number Generation function, a random set of normally distributed data (45 numbers) was generated using the sample mean and the sample standard. The last three digits of my student number (403) were used as the random seed variable. The Excel data analysis tool was used to analyze the random data, and the sample mean, sample variance and standard deviation, standard error, and the 90% confidence interval for the population mean were calculated.
RESULTS & DISCUSSION Table 1, Table 2 and the equation sheet in the appendices show the results of the statistical analysis. All the results are in one thousandth of an inch. For the knob measurements, the sample mean was 141.6 with a variance of 14.29 and a standard deviation of 3.780, the standard error was .05634 and the population mean with 90% confidence interval was .9467. Figure 1 shows the histogram of the frequency of the knob size. The histogram is skewed to the left with the greatest frequency in the 143 to 145 category. If one sample had been recorded as 0.114", I would say there is a problem either with the machine or the measuring device. The statistics determined for the sample data with the two methods (the equations vs. Excel’s data tools) were the same. The statistics determined for the data that were randomly generated, and the statistics determined for the knob measurements were different. For the random set of normally distributed data, the sample mean was 141.2, sample variance was 18.08, sample standard deviation was 4.252, standard error was .6338, and the 90% confidence interval for the population mean was 1.065. The proportional mean value was 0.60 ± 0.14 with 95% confidence. On the assumed Gaussian distribution the 99% confidence interval shows the mean value was 141.6 ±1.451.
CONCLUSIONS The objective of this lab was fulfilled by analyzing the production data. The probability of producing a knob in the acceptable range was calculated as 60%±14%. Therefore it can be predicted with 99% certainty that 60% ± 14% of the knobs produced will be in the acceptable range of 0.137" to 0.143". In other words I would predict that 27 (give or take 6) out of 45 knobs would be the correct size. 2
Zhang, Xueli
PPENDICES- FIGURES AND TABLES Table 1. The Frequency Table for the Histogram Lower
upper
Class
frequency
131
133 131 up to133
1
133
135 133 up to135
2
135
137 135 up to137
1
137
139 137 up to139
6
139
141 139 up to141
6
141
143 141 up to143
8
143
145 143 up to145
9
145
147 145 up to147
7
147
149 147 up to149
5
Total
45
Figure 1. The Histogram of Knob Size versus Frequency
Knob Size Versus Frequency 10 F 9 r 8 e 7 q 6 u 5 e 4 n 3 c 2 y 1 0 131 up 133 up 135 up 137 up 139 up 141 up 143 up 145 up 147 up to133 to135 to137 to139 to141 to143 to145 to147 to149 Knob Size
3
Zhang, Xueli Table 2. Summary Table Sample Statistic
Result via Equations(thou)
Result via MS Excel (thou)
Sample Mean Sample Variance Sample Standard Deviation Standard Error The 90% Confidence Interval for the Population Mean
141.6 14.29 3.780 0.5634 0.9467
141.6 14.29 3.780 0.5634 0.9467
4
Random Normal Distribution using Excel with random seed variable (thou) 141.2 18.08 4.252 0.6338 1.065
