Six Steps for OneSample Proportion Hypothesis Test
Steps 13 Section
Let's apply the general steps for hypothesis testing to the specific case of testing a onesample proportion.
 Step 1: Set up the hypotheses and check conditions.

\( np_0\ge 5 \) and \(n(1−p_0)≥5 \)
One Proportion Ztest Hypotheses
LeftTailed \( H_0\colon p=p_0 \)
 \( H_a\colon p<p_0\)
RightTailed \( H_0\colon p=p_0 \)
 \( H_a\colon p>p_0 \)
TwoTailed \( H_0\colon p=p_0 \)
 \( H_a\colon p\ne p_0 \)
 Step 2: Decide on the level of significance \(\boldsymbol{(\alpha)}\).
 Step 3: Calculate the test statistic.

One Proportion Ztest: \(z^*=\dfrac{\hat{p}p_0}{\sqrt{\frac{p_0(1p_0)}{n}}} \)
Rejection Region Approach
Steps 46 Section
 Step 4: Find the appropriate critical values for the tests. Write down clearly the rejection region for the problem.

LeftTailed Test
RightTailed Test
TwoTailed Test
View the critical values and regions with an \(\alpha=.05\).
 Step 5: Make a decision about the null hypothesis.
 Check to see if the value of the test statistic falls in the rejection region. If it does, then reject \(H_0 \) (and conclude \(H_a \)). If it does not fall in the rejection region, do not reject \(H_0 \).
 Step 6: State an overall conclusion.
PValue Approach
Steps 46 Section
 Step 4: Compute the appropriate pvalue based on our alternative hypothesis.

LeftTailed
 \(P(Z \le z^*)\)
RightTailed \(P(Z\ge z^*)\)
TwoTailed \(2\) x \(P(Z \ge z^*)\)
 Step 5: Make a decision about the null hypotheses.
 If the pvalue is less than the significance level, then reject the null hypothesis. If the pvalue is greater than the significance level, fail to reject the null hypothesis.
 Step 6: State an overall conclusion.
Example 65: Penn State Students from Pennsylvania Section
Referring back to example 64. 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?
Conduct the test using both the rejection region and pvalue approach.
 Step 1: Set up the hypotheses and check conditions.

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:
\( H_0\colon p=0.5 \) vs \(H_a\colon p>0.5 \)
Can we use the ztest statistic? The answer is yes since the hypothesized value \(p_0 \) is \(0.5\) and we can check that: \(np_0=500(0.5)=250 \ge 5 \) and \(n(1p_0)=500(10.5)=250 \ge 5 \)
 Step 2: Decide on the significance level, \(\alpha \).

According to the question, \(\alpha= 0.05 \).
 Step 3: Calculate the test statistic:

\begin{align} z^*&= \dfrac{0.5560.5}{\sqrt{\frac{0.5(10.5)}{500}}}\\z^*&=2.504 \end{align}
Rejection Region Approach
 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 the standard normal table to find the value of \(Z_{0.05} \). From the table, \(Z_{0.05} \) is found to be \(1.645\) and thus the critical value is \(1.645\). The rejection region for the righttailed test is given by:
\( z^*>1.645 \)
 Step 5: Make a decision about the null hypothesis.

The test statistic or the observed Zvalue is \(2.504\). Since \(z^*\) falls within the rejection region, we reject \(H_0 \).
 Step 6: State an overall conclusion.

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.
PValue Approach
 Step 4: Compute the appropriate pvalue based on our alternative hypothesis:
 \(\text{pvalue}=P(Z\ge z^*)=P(Z \ge 2.504)=0.0062\)
 Step 5: Make a decision about the null hypothesis.

Since \(\text{pvalue} = 0.0062 \le 0.05\) (the \(\alpha \) value), we reject the null hypothesis.
 Step 6: State an overall conclusion.

With a test statistic of \(2.504\) and pvalue 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.
Try it!
Online Purchases Section
An ecommerce research company claims that 60% or more graduate students have bought merchandise online. A consumer group is suspicious of the claim and thinks that the proportion is lower than 60%. A random sample of 80 graduate students shows that only 22 students have ever done so. Is there enough evidence to show that the true proportion is lower than 60%?
Conduct the test at 10% Type I error rate and use the pvalue and rejection region approaches.
 Step 1: Set up the hypotheses and check conditions.

Set up the hypotheses. Since the research hypothesis is to check whether the proportion is less than 0.6 we set it up as a one (left)tailed test:
\( H_0\colon p=0.6 \) vs \(H_a\colon p<0.6 \)
Can we use the ztest statistic? The answer is yes since the hypothesized value \(p_0 \) is 0.6 and we can check that: \(np_0=80(0.6)=48 \ge 5 \) and \(n(1p_0)=80(10.6)=32 \ge 5 \)
 Step 2: Decide on the significance level, \(\alpha \).

According to the question, \(\alpha= 0.1 \).
 Step 3: Calculate the test statistic:

\begin{align} z^* &=\frac{\hat{p}p_0}{\sqrt{\frac{p_0(1p_0)}{n}}}\\&=\frac{.2750.6}{\sqrt{\frac{0.6(10.6)}{80}}}\\&=5.93 \end{align}
Rejection Region Approach
 Step 4: Find the appropriate critical values for the test using the ztable. Write down clearly the rejection region for the problem.

The critical value is the value of the standard normal where 10% fall below it. Using the standard normal table, we can see that the value is 1.28.
 Step 5: Make a decision about the null hypothesis.

The rejection region is any \(z^* \) such that \(z^*<1.28 \) . Since our test statistic, 5.93, is inside the rejection region, we reject the null hypothesis.
 Step 6: State an overall conclusion.

There is enough evidence in the data provided to suggest, at 10% level of significance, that the true proportion of students who made purchases online was less than 60%.
PValue Approach
 Step 4: Compute the appropriate pvalue based on our alternative hypothesis:
 \( \text{pvalue}=P(Z \le 5.93) = 0.0000000003 \)
 Step 5: Make a decision about the null hypothesis.

Since our pvalue is very small and less than our significance level of 10%, we reject the null hypothesis.
 Step 6: State an overall conclusion.

There is enough evidence in the data provided to suggest, at 10% level of significance, that the true proportion of students who made purchases online was less than 60%.