Question

The supervisor of a production line that assembles computer keyboards has been experiencing problems since a new process was instituted. He notes that there has been an increase in defective units and occasional backlogs when one station’s productivity is not matched by that of other stations. Upon reviewing the process, the supervisor discovered that the management scientists who developed the production process assumed that the amount of time to complete a critical part of the process is normally distributed with a mean of 130 seconds and a standard deviation of 15 seconds. He is satisfied that the process time is normally distributed with a standard deviation of 15 seconds but he is unsure about the mean time. In order to examine the problem he measures the time of 100 assemblies. The mean of these times was calculated to be 126.8 seconds. Can the supervisor conclude at the 5% significance level that the assumption that the mean assembly time if 130 seconds is incorrect?

1. State the null and alternative hypothesis that best tests the hypothesis of interest. Be sure to pay attention to whether this should be a 1-tailed or a 2-tailed test.
2. Carry out this test, and state whether you reject or fail to reject the null hypothesis. In computing the test statistic, be sure to indicate whether you are using “t” or “z”. Use the precise critical value of the “t” or “z” you use, not an approximation.  

Answer 1

Answer a:
According to given,

The Null Hypothesis, H0: = 130 seconds and

The Alternate Hypothesis, Ha: ≠ 130 seconds (where is the mean assembly time)

It is a 2 - Sided Test

Answer b:
It is already given that the population process time is normally distributed with parameters, mean = 130 seconds and standard deviation = 15 seconds

The supervisor is satisfied that the standard deviation for the process time = 15 seconds

So, for the above test the Standard Deviation is a known measure.

Also given, the mean process time of a sample of 100 assemblies = 126.8 seconds

Let X represent a process time in the sample of 100

Sample Size, n = 100, Sample Mean Time, = 126.8 seconds and Standard Deviation, = 15 seconds

The test statistic for the above test, Z = [(n^0.5) ( - )] / which follows Standard Normal Distribution under the null hypothesis

Substituting all values,

The value of test statistic, Z = -2.13

Given Significance Level, = 5% = 0.05,   / 2 = 0.025

Since, it is a 2 - sided test, we have to use   / 2 value to find the critical value.

The Z - Score corresponding to 0.025, Z = -1.96

In general, if the numerical value of the test statistic exceeds the critical value we reject the null hypothesis, else we fail to reject the null.

In this case, The numerical value of test statistic > Critical value

So, the null hypothesis is rejected  

The Conclusion: There is enough evidence to support the claim that the mean assembly line is ≠ 130 seconds