Hypothesis testing the mean when σ is known
Let's see how Seller Door would conduct the hypothesis test, assuming the population standard deviation is known to be σ = 24 seconds. Let's follow the step-by-step guide from Section 8.3!
State the hypotheses. The company starts by writing down the null and alternative hypotheses that are being included in the test. This makes it clear what is being tested and whether the test is one-sided or two-sided. In this test, the company is testing whether the average sales routine time has changed from the old value of 620 seconds:
H0: μ = 620
HA: μ ≠ 620
Assume the null hypothesis is true. The next step in the test is to assume that the null hypothesis is true. In this case, this means assuming that the population mean sales routine time is 620 seconds. This step doesn't require a lot of work, but it is important because this assumption is maintained for the rest of the test (until it is possibly rejected)! It is a good idea to write the assumption down underneath the two hypotheses: 'Assume μ = 620'.
Choose a level of significance, α. Throughout this chapter we have discussed the importance of the level of significance. The company can choose any level for α between 0 and 1, but this level will indicate how strong the evidence needs to be before the company rejects the null hypothesis. A low level of significance means that the company requires a lot of evidence.
Let's suppose the company chooses α = 0.05, which is a very common level of significance. As we are about to see, and as we have discussed before, the impact of the level of significance is that it determines the critical values and the region of rejection.
Determine the critical value(s). Once the level of significance is chosen, the critical values can be calculated. As explained in the step-by-step guide, since this test is a two-sided test, there will be two critical values: zα/2 and -zα/2. Because the level of significance is α = 0.05, the critical values are z0.025 and -z0.025.
The standard normal table or appropriate statistical software can be used to find these two z-scores. They are z0.025 = 1.96 and -z0.025 = -1.96.
Determine the region of rejection. The critical values specify a region of rejection in the standard normal distribution, as outlined in the step-by-step guide. Since there are two critical values, the region of rejection is the region outside of them: the set of values greater than 1.96 and values less than -1.96.
Collect a sample and calculate a sample statistic. As we mentioned earlier, Seller Door intended to collect a sample of 36 time values. Let's suppose that this sample has been collected and a sample mean of x = 628.8 seconds was calculated.
Calculate the test statistic. The test statistic is calculated from the sample mean using the appropriate transformation formula. Assuming the null hypothesis is true, the sampling distribution of the mean, X, follows the normal distribution with mean μ0 = 620 and standard deviation σ/√n = 24/√36 = 4. So the z-score test statistic of the sample mean of 627.4 is:
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| = | 2.20 |
Conclusion. Seller Door is now in a position to conclude the hypothesis test. As outlined in the step-by-step guide, the conclusion rests on whether or not the test statistic is in the region of rejection.
The test statistic of 2.20 is greater than 1.96 and therefore is in the region of rejection. Therefore, Seller Door rejects the null hypothesis. There is enough evidence to conclude that the average sales routine time has changed.