Question

How is the rejection region defined, and how is that related to the p value? When do you reject or fail to reject the null hypothesis? Why do you think statisticians are asked to complete hypothesis testing? Can you think of examples in courts, in medicine, or in your area?

Answer 1

Answer:

Two Methods for Making a Statistical Decision

There are two approaches to making a statistical decision regarding a null hypothesis. One is the rejection region approach and the second is the p-value (or probability value) approach. Of the two methods, the latter is more commonly used and provided in the published literature. However, understanding the rejection region approach can go a long way in one's understanding of the p-value method. Regardless of a method applied, the conclusions from the two approaches are exactly the same.

Test statistic: The sample statistic one uses to either reject H_o (and conclude H_a) or not to reject.

Critical values: The values of the test statistic that separate the rejection and non-rejection regions.

Rejection region: the set of values for the test statistic that leads to rejection of H_o.

Non-rejection region: the set of values not in the rejection region that leads to non-rejection of.

P-value: The p-value (or probability value) is the probability that the test statistic equals the observed value or a more extreme value under the assumption that the null hypothesis is true.

Steps in a Conducting a Hypothesis Test

Step 1. Check the conditions necessary to run the selected test and select the hypotheses for that test.:

1. If Z-test for one proportion: np0?5 and n(1?p0)?5
2. If a t-test for one mean: either the data comes from an approximately normal distribution or the sample size is at least 30. If neither, then the data is not heavily skewed and without outliers.

Step 2. Decide on the significance level, ?.

Step 3. Compute the value of the test statistic:

Step 4. Find the appropriate critical values for the tests using the Z-table for a test of one proportion, or the t-table if a test for one means. REMEMBER: for the one mean test the degrees for freedom are the sample size minus one (i.e. n - 1). Write down clearly the rejection region for the problem

Step 5. Check to see if the value of the test statistic falls in the rejection region. If it does, then reject H0 (and conclude Ha). If it does not fall in the rejection region, do not reject H0.

Step 6. State the conclusion in words.

P-value Approach to Hypothesis Testing

Steps 1- Step 3. The first few steps (Step 0 - Step 3) are exactly the same as the rejection region approach.

Step 4. In Step 4, we need to compute the appropriate p-value based on our alternative hypothesis:

If HaHa is right-tailed, then the p-value is the probability the sample data produces a value equal to or greater than the observed test statistic.

If HaHa is left-tailed, then the p-value is the probability the sample data produces a value equal to or less than the observed test statistic.

If Ha is two-tailed, then the p-value is two times the probability the sample data produces a value equal to or greater than the absolute value of the observed test statistic.

Step 5. Check to see if the p-value is less than the stated alpha value. If it is, then reject H0 (and conclude Ha). If it is not less than alpha, do not reject H0.

Step 6. A conclusion in words.

Example: Penn State Students from Pennsylvania

Continuing with our one-proportion example at the beginning of this lesson, say we take a random sample of 500 Penn State students and find that 278 are from Pennsylvania. Can we conclude that the proportion is larger than 0.5 at a 5% level of significance?

A: Using the Rejection Region Approach

Step 1. Can we use the one-proportion z-test?

The answer is yes since the hypothesized value p0 is 0.5 and we can check that:

np0=500×0.5=250?5
n(1?p0)=500×(1?0.5)=250?5

Set up the hypotheses. Since the research hypothesis is to check whether the proportion is greater than 0.5 we set it up as a one(right)-tailed test:

H0:p=0.5 Ha:p>0.5

Step 2. Decide on the significance level, ?.

According to the question, ?? = 0.05.

Step 3. Compute the value of the test statistic:

Z* = (p - p0) / sqrt(PQ/n) = 2.504

Step 4. Find the appropriate critical values for the test using the z-table. Write down clearly the rejection region for the problem. We can use the standard normal table or the last row of our t-table to find the value of Z_0.05 since that last row for df = ?(infinite) refers to the z-value.

From the table, Z0.05 is found to be 1.645 and thus the critical value is 1.645. The rejection region for the right-tailed test is given by:

Z*>1.645

Step 5. Check whether the value of the test statistic falls in the rejection region. If it does, then reject H0 (and conclude Ha). If it does not fall in the rejection region, do not reject H0.

The observed Z-value is 2.504 - this is our test statistic. Since Z* falls within the rejection region, we rejectH0.

Step 6. State the conclusion in words.

With a test statistic of 2.504 and critical value of 1.645 at a 5% level of significance, we have enough statistical evidence to reject the null hypothesis. We conclude that a majority of the students are from Pennsylvania.

Using the P-value Approach

Steps 1- Step 3. The first few steps (Step 1 - Step 3) are exactly the same as the rejection region approach.

Step 4. In Step 4, we need to compute the appropriate p-value based on our alternative hypothesis. With our alternative hypothesis being right-tailed:

P value = P(Z> Z*) = 0.0062

Step 5. Since p-value = 0.0062 < 0.05 (the ? value), we reject the null hypothesis.

Step 6. A conclusion in words:

With a test statistic of 2.504 and p-value of 0.0062, we reject the null hypothesis at a 5% level of significance. We conclude that a majority of the students are from Pennsylvania.