Unit Summary 

Reading Assignment
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 pvalue 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:
 Null and alternative hypotheses
 Level of significance
 Test statistics
 Critical values and rejection region
 Process of checking to see whether the test statistic falls in the rejection region
 Conclusion in words
Test statistic: The sample statistic one uses to either reject H_{o} (and conclude H_{a}) or not to reject H_{o}.
Critical values: The values of the test statistic that separate the rejection and nonrejection regions.
Rejection region: the set of values for the test statistic that leads to rejection of H_{o}.
Nonrejection region: the set of values not in the rejection region that leads to nonrejection of H_{o}.
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.
Oneproportion Ztest for
Step 0. Check the conditions for the oneproportion ztest to be valid:
 n_{o} 5
 n(1  _{o}) 5
Step 1. Set up the hypotheses as one of:
Twotailed Righttailed Lefttailed H_{o}: = _{o} ORH_{o}: = _{o} ORH_{o}: = _{o} H_{a}: _{o} H_{a}: > _{o} H_{a}: < _{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 ztable. 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 H_{o} (and conclude H_{a}). If it does not fall in the rejection region, do not reject H_{o}.
Step 6. State the conclusion in words.
Some expert claims that the probability of each person being lefthanded is 0.25. It is observed that out of 30 randomly sampled people, 10 are lefthanded. 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 oneproportion ztest?
The answer is yes since the hypothesized value _{o} is 0.25 and we can check that:
n_{o} = 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 twotailed test):
H_{o}: = 0.25
H_{a}: 0.25Step 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 ztable. Write down clearly the rejection region for the problem. We can use Table 2 to find the value of Z_{0.025} since the row for df = (infinite) refers to the zvalue.
From Table 2, Z_{0.025} is found to be 1.96 and thus the critical values are ± 1.96. The rejection region for the twotailed 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 H_{o} (and conclude H_{a}). If it does not fall in the rejection region, do not reject H_{o}.
The observed zvalue is 1.05 and will be denoted as z*. Since z* does not fall within the rejection region, we do not reject H_{o}.
Step 6. State the conclusion in words.
Based on the observed data, there is not enough evidence to conclude that the population proportion of lefthanded people is different from 0.25.
b. Use the pvalue 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 pvalue. Since it is a twotailed test:
Step 5. Since pvalue = 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 lefthanded people is different from 0.25.
Comparing the PValue Approach to the Rejection Region Approach
Both approaches will ensure the same conclusion and either one will work. However, using the pvalue approach has the following advantages:
 Using the rejection region approach, you need to check the table for the critical value every time people give you a different value.
 In addition to just using it to reject or not reject H_{o} by comparing pvalue to value, pvalue also gives us some idea of the strength of the evidence against H_{o}.