Case Summary

Quality Associates, Inc., a consulting firm, advises its clients about sampling and statistical procedures that can be used to control their manufacturing processes. In one particular application, a client gave Quality Associates a sample of 800 observations taken during a time in which that client’s process was operating satisfactorily.

Methodology

Quality Associates then suggested that random samples of size 30 be taken periodically to monitor the process on an ongoing basis. By analyzing the new samples, the client could quickly learn whether the process was operating satisfactorily. When the process was not operating satisfactorily, corrective action could be taken to eliminate the problem.

Assumptions

The sample standard deviation for these data was 0.21. Given the large dataset, the population standard deviation was assumed to be 0.21. The design specification indicated the mean for the process should be 12.

Hypotheses

The hypothesis test suggested by Quality Associates is:

• \(H_0: \mu = 12\)

• \(H_a: \mu \ne 12\)

Corrective action will be taken any time H0 is rejected.

Data

Samples were collected at hourly intervals during the first day of operation of the new statistical process control procedure. These data are available in the data set Quality.

Hypothesis Test

We conducted a hypothesis test for each sample at the 0.01 level of significance to determine if action was needed.

• Since n >= 30, we can approximate the sampling distribution as roughly normal.

Sample statistics at the 1% significance level

##      Mean z_value p_value h0_fail_to_reject
## 1 11.9587 -1.0772  0.2814              TRUE
## 2 12.0287  0.7486  0.4541              TRUE
## 3 11.8890 -2.8951  0.0038             FALSE
## 4 12.0813  2.1205  0.0340              TRUE

Conclusions

For each sample, the p-value is the probablity of obtaining a sample with a mean this far away from 12, given that it is drawn from a population with a mean of 12. To run the hypothesis test, p-values were compared to the significance level of 1%. In three of the 4 samples, p-values were greater then this significance level and we therefore failed to reject H0. In the case of sample 3, the probability of obtaining a mean this extreme is lower then 1%. Therefore, sample 3 failed the hypothesis test for H0 at a significance level of 1% and corrective action must be taken.

Standard Deviation

We computed the standard deviation for each of the four samples and reviewed whether the assumption of 0.21 for the population standard deviation appeared reasonable.

Standard deviations for the four samples:

## [1] 0.2204 0.2204 0.2072 0.2061

Average of standard deviations:

## [1] 0.2135

Conclusions

Yes, the assumption of a population standard deviation of 21% seems reasonable.

Confidence Interval

We computed limits for the sample mean x around m = 12 such that, as long as a new sample mean is within those limits, the process will be considered to be operating satisfactorily. If x exceeds the upper limit or if x is below the lower limit, corrective action will be taken. These limits are referred to as upper and lower control limits for quality control purposes.

## [1] "At a significance level of 1% the lower control limit is 11.9012 and the upper is 12.0988"

Comparing these limits to the sample means, we see that the third sample is beneath the lower limit. This confirms our hypothesis test’s conclusion that corrective action is needed. (As we would expect, given that we used the same significance level for both calculations.)

## [1] 11.9587 12.0287 11.8890 12.0813

Significance

Quality Associates chose a significance level of 1%, determining the level of risk of Type A errors that is considered acceptable. This significance level means that if we took 100 samples from a population with the mean of 12, we expect only one of them to have a mean far enough from 12 that we would incorrectly flag it as as problem.

By setting the significance level low, we are minimizing the risk of unnecessary repairs. If we increase the significance level, we will flag more samples as problematic. In other words, we will increase the risk of making unnessary repairs.

To demonstrate, let’s compare our current hypothesis test at 1% significance to a new hypothesis test with a 5% significance level. In this table, “FALSE” means that H0 was rejected and corrective action must be taken.

##   h0_fail_to_reject h0_5_percent_significance
## 1              TRUE                      TRUE
## 2              TRUE                      TRUE
## 3             FALSE                     FALSE
## 4              TRUE                     FALSE

As expected, increasing the significance level increases the number of times H0 will be rejected and we will take corrective action.