The p-Value Approach

The p-value approach to hypothesis testing is very similar to the critical value approach (see previous post). Rather than deciding whether or not to reject the null hypothesis based on whether the test statistic falls in a rejection region or not, the p-value approach allows us to make the decision based on whether or not the p-value of the sample data is more or less than the level of confidence.

The p-value is the probability of getting a test statistic equal to or more extreme than the sample result. If the p-value is greater than the level of confidence then we can say that the probability of a more extreme test statistic is larger than the level of confidence and thus we do not reject H_{0}.

If, on the other hand, the p-value is less than the level of confidence, we conclude that the probability of a more extreme test statistic is smaller than the level of confidence and thus we reject H_{0}.

The five step methodology of the p-value approach to hypothesis testing is as follows:

(Note: The first three steps are identical to the critical value approach described in the previous post. However, step 4, the calculation of the critical value, is omitted in this method. Differences in the final two steps between the critical value approach and the p-value approach are emphasized.)

State the Hypotheses

1. State the null hypothesis, H_{0}, and the alternative hypothesis, H_{1}.

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.

Conduct the Study

4. Collect the data and compute the test statistic and the p-value.

Draw Conclusions

5. Evaluate the p-value 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 (we'll look at the same example as the last post, also 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.H_{0}: μ ≤ 52

H_{1}: 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.

The Study

4. 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. Next, we look up the p-value of 0.88. The cumulative normal distribution table tells us that the area to the left of 0.88 is 0.8106. Therefore, the p-value of 0.88 = 1-0.8106 = 0.1894.

Conclusions

5. Since 0.1894 is greater than the level of significance, α, 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.

## Sunday, February 24, 2008

### Hypothesis Testing - p-Value Approach - 5 Step Methodology

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