6a.4.3 - Steps in Conducting a Hypothesis Test for \(p\)

Six Steps for One-Sample Proportion Hypothesis Test

Steps 1-3 Section

Let's apply the general steps for hypothesis testing to the specific case of testing a one-sample proportion.

Step 1: Set up the hypotheses and check conditions.

\( np_0\ge 5 \) and \(n(1−p_0)≥5 \)

One Proportion Z-test Hypotheses

Left-Tailed
\( H_0\colon p=p_0 \)
\( H_a\colon p<p_0\)
Right-Tailed
\( H_0\colon p=p_0 \)
\( H_a\colon p>p_0 \)
Two-Tailed
\( H_0\colon p=p_0 \)
\( H_a\colon p\ne p_0 \)

Step 2: Decide on the level of significance \(\boldsymbol{(\alpha)}\).

Step 3: Calculate the test statistic.

One Proportion Z-test: \(z^*=\dfrac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \)

The first few steps (Step 1 - Step 3) are exactly the same as the rejection region or p-value approach. The next part will discuss steps 4 - 6 for both approaches. 

Rejection Region Approach

Steps 4-6 Section

Step 4: Find the appropriate critical values for the tests. Write down clearly the rejection region for the problem.

 

Left-Tailed Test

α z α

Reject \(H_0\) if \(z^* \le z_\alpha\)


Right-Tailed Test

  α 1 - α z 1-α

Reject \(H_0\) if \(z^* \ge z_{1-\alpha}\)


Two-Tailed Test

α/2 1 - α α/2 z 1-α\2 z α/2

Reject \(H_0\) if \(|z^*| \ge |z_{\alpha/2}|\)

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.

P-Value Approach

Steps 4-6 Section

Step 4: Compute the appropriate p-value based on our alternative hypothesis.
Left-Tailed
\(P(Z \le z^*)\)
Right-Tailed
\(P(Z\ge z^*)\)
Two-Tailed
\(2\) x \(P(Z \ge |z^*|)\)
Step 5: Make a decision about the null hypotheses.
If the p-value is less than the significance level, then reject the null hypothesis. If the p-value is greater than the significance level, fail to reject the null hypothesis.
Step 6: State an overall conclusion.
Note! Recall that the P-value is a probability of obtaining a value of the test statistic or a more extreme value of the test statistic assuming that the null hypothesis is true.

Example 6-5: Penn State Students from Pennsylvania Section

Flag of Pennsylvania

Referring back to example 6-4. 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 p-value approach.

Answer
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 z-test 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 \(n(1-p_0)=500(1-0.5)=250 \ge 5 \)

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{0.556-0.5}{\sqrt{\frac{0.5(1-0.5)}{500}}}\\z^*&=2.504 \end{align}

Rejection Region Approach

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 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 right-tailed test is given by:

\( z^*>1.645 \)

Step 5: Make a decision about the null hypothesis.

The test statistic or the observed Z-value 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.

P-Value Approach

 
Step 4: Compute the appropriate p-value based on our alternative hypothesis:
\(\text{p-value}=P(Z\ge z^*)=P(Z \ge 2.504)=0.0062\)

Step 5: Make a decision about the null hypothesis.

Since \(\text{p-value} = 0.0062 \le 0.05\) (the \(\alpha \) value), we reject the null hypothesis.

Step 6: State an overall conclusion.

With a test statistic of \(2.504\) and 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.

Try it!

Online Purchases Section

An e-commerce 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 p-value 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 z-test 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 \(n(1-p_0)=80(1-0.6)=32 \ge 5 \)

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(1-p_0)}{n}}}\\&=\frac{.275-0.6}{\sqrt{\frac{0.6(1-0.6)}{80}}}\\&=-5.93 \end{align}

Rejection Region Approach

Step 4: Find the appropriate critical values for the test using the z-table. 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%.

P-Value Approach

Step 4: Compute the appropriate p-value based on our alternative hypothesis:
\( \text{p-value}=P(Z \le -5.93) = 0.0000000003 \)
Step 5: Make a decision about the null hypothesis.

Since our p-value 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%.