A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next. The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the computed value of t?
A)+2.797
B)-1.000
C)-3.299
D)0.5938

Question

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the computed value of t?
A)+2.797 
B)-1.000 
C)-3.299 
D)0.5938

Quizplus · Accepted Answer

To compute the pooled standard deviation and the t statistic, we use the formula:

$$ s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}} $$

where $n_1$ and $n_2$ are the sample sizes, $s_1$ and $s_2$ are the sample standard deviations, and $s_p$ is the pooled standard deviation. Then the t statistic is:

$$ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} $$

where $\bar{x}_1$ and $\bar{x}_2$ are the sample means.

Plugging in the values from the table, we get:

$$ s_p = \sqrt{\frac{(15-1)0.31^2 + (11-1)0.35^2}{15+11-2}} \approx 0.3299 $$

$$ t = \frac{0.27 - 0.34}{0.3299 \sqrt{\frac{1}{15}+\frac{1}{11}}} \approx -3.299 $$

Therefore, the computed value of t is -3.299, which corresponds to choice C.

A National Manufacturer of Ball Bearings Is Experimenting with Two