# Lesson 9.1 - Statistical Test Using Rejection Region Approach

 Unit Summary Rejection Region Approach to Hypothesis Testing for One Proportion Problem Comparing the P-Value Approach to the Rejection Region Approach

An Introduction to Statistical Methods and Data Analysis, chapters 10.2, 5.6 and 5.7.

Rejection Region Approach to Hypothesis Testing for One Proportion Problem

One can perform hypothesis testing using the p-value approach, or one can perform hypothesis testing using a rejection region approach. The conclusions from the two approaches are exactly the same.

There are six parts of a test when using the rejection region approach:

1. Null and alternative hypotheses
2. Level of significance
3. Test statistics
4. Critical values and rejection region
5. Process of checking to see whether the test statistic falls in the rejection region
6. Conclusion in words

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

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 Ho.

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

As mentioned in lesson 8, the logic of hypothesis testing is to reject the null hypothesis if the sample data are not consistent with the null hypothesis. Thus, one rejects the null hypothesis if the observed test statistic is more extreme in the direction of the alternative hypothesis than one can tolerate. The critical values are the boundary values obtained corresponding to the preset level.

One-proportion Z-test for

Step 0. Check the conditions for the one-proportion z-test to be valid:

1. no 5
2. n(1 - o) 5

Step 1. Set up the hypotheses as one of:

 Two-tailed Right-tailed Left-tailed Ho: = o OR Ho: = o OR Ho: = o Ha: o Ha: > o Ha: < o

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. 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 Ho (and conclude Ha). If it does not fall in the rejection region, do not reject Ho.

Step 6. State the conclusion in words.

Some expert claims that the probability of each person being left-handed is 0.25. It is observed that out of 30 randomly sampled people, 10 are left-handed. Using = 0.05, is there sufficient evidence to conclude that the population proportion is different from 0.25?

a. Use the rejection region approach to perform the testing.

Step 0. Can we use the one-proportion z-test?

The answer is yes since the hypothesized value o is 0.25 and we can check that:

no = 30 · 0.25 = 7.5 5,
n(1 - o) = 30 · (1 - 0.25) = 22.5 5.

Step 1. Set up the hypotheses (since the research hypothesis is to check whether the proportion is different from 0.25, we set it up as a two-tailed test):

Ho: = 0.25
Ha: 0.25

Step 2. Decide on the significance level, .

According to the question, = 0.05.

Step 3. Compute the value of the test statistic:

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 Table 2 to find the value of Z0.025 since the row for df = (infinite) refers to the z-value.

From Table 2, Z0.025 is found to be 1.96 and thus the critical values are ± 1.96. The rejection region for the two-tailed test is given by:

z > 1.96 or z < -1.96

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

The observed z-value is 1.05 and will be denoted as z*. Since z* does not fall within the rejection region, we do not reject Ho.

Step 6. State the conclusion in words.

Based on the observed data, there is not enough evidence to conclude that the population proportion of left-handed people is different from 0.25.

b. Use the p-value approach to perform the testing.

Step 0 - 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 p-value. Since it is a two-tailed test:

Step 5. Since p-value = 0.2938 > 0.05 (the value), we cannot reject the null hypothesis.

Step 6. Conclusion in words:

Based on the observed data, there is insufficient evidence to conclude that the population proportion of left-handed people is different from 0.25.

Comparing the P-Value Approach to the Rejection Region Approach

Both approaches will ensure the same conclusion and either one will work. However, using the p-value approach has the following advantages:

1. Using the rejection region approach, you need to check the table for the critical value every time people give you a different value.
2. In addition to just using it to reject or not reject Ho by comparing p-value to value, p-value also gives us some idea of the strength of the evidence against Ho.