Sunday, February 24, 2008

Hypothesis Testing - Critical Value Approach - 6 Step Methodology

The six-step methodology of the Critical Value Approach to hypothesis testing is as follows:
(Note: The methodology below works equally well for both one-tail and two-tail hypothesis testing.)

State the Hypotheses
1. State the null hypothesis, H0, and the alternative hypothesis, H1.
Design the Study
2. Choose the level of significance, α according to the importance of the risk or committing Type I errors. Determine the sample size, n, based on the resources available to collect the data.
3. Determine the test statistic and sampling distribution. When the hypotheses involve the population mean, μ, the test statistic is z when σ is known and t when σ is not known. These test statistics follow the normal distribution and the t-distribution respectively.
4. Determine the critical values that divide the rejection and non-rejection regions.
Note: For ethical reasons, the level of significance and critical values should be determined prior to conducting the test. The test should be designed so that the predetermined values do not influence the test results.
Conduct the Study
5. Collect the data and compute the test statistic.
Draw Conclusions
6. Evaluate the test statistic and determine whether or not to reject the null hypothesis. Summarize the results and state a managerial conclusion in the context of the problem.

Example (reviewed at the beginning of Lecture 7):
A phone industry manager thinks that customer monthly cell phone bills have increased and now average over $52 per month. The company asks you to test this claim. The population standard deviation, σ, is known to be equal to 10 from historical data.

The Hypotheses
1.H0: μ ≤ 52
H1: μ > 52
Study Design
2. After consulting with the manager and discussing error risk, we choose a level of significance, α, of 0.10. Our resources allow us to sample 64 sample cell phone bills.
3. Since our hypothesis involves the population mean and we know the population standard deviation, our test statistic is z and follows the normal distribution.
4. In determining the critical value, we first recognize this test as a one-tail test since the null hypothesis involves an inequality, ≤. Therefore the rejection region is entirely on the side of the distribution greater than the historic mean - right tail.
We want to determine a z-value for which the area to the right of that value is 0.10, our α. We can use the cumulative normal distribution table (which gives areas to the left of the z-value) and find z having value 0.90 = 1.285. This is our critical value.
The Study
5. We conduct our study and find that the mean of the 64 sample cell phone bills is 53.1. We compute the test statstic, z = (xbar-μ)/(σ/√n) = (53.1-52)/(10/√64) = 0.88.
6. Since 0.88 is less than the critical value of 1.285, we do not reject the null hypothesis. We report to the company that, based on our testing, there is not evidence that the mean cell phone bill has increased from $52 per month.

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