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Intervals & Tests
The following is an excerpt from The Quality Engineering Handbook by Thomas Pyzdek, © QA Publishing, LLC.
Experiment: The nominal specification for filling a bottle with a test chemical is 30 cc. The plan is to draw a sample of n=25 units from a stable process and, using the sample mean and standard deviation, construct a two-sided confidence interval (an interval that extends on either side of the sample average) that has a 95% probability of including the true population mean. If the inter-val includes 30, conclude that the lot mean is 30, otherwise conclude that the lot mean is not 30.
Result: A sample of 25 bottles was measured and the following statistics computed
The appropriate test statistic is t, given by the formula
Table 6 in the Appendix gives values for the t statistic at various degrees of freedom. There are n-1 degrees of freedom. For our example we need the t.975 column and the row for 24 df. This gives a t value of 2.064. Since the absolute value of this t value is greater than our test statistic, we fail to reject the hypothesis that the lot mean is 30 cc. Using statistical notation this is shown as:
H0: m = 30 cc (the null hypothesis)
H1: m is not equal to 30 cc (the alternate hypothesis)
α = .05 (type I error or level of significance)
Critical region: -2.064 ² t0 ² +2.064
Test statistic: t = -1.67.
Since t lies inside the critical region, fail to reject H0, and accept the hypothesis that the lot mean is 30cc for the data at hand.