Chapter 8, Problem 8CQ

a.

To determine

Find the value of test statistic.

Answer to Problem 8CQ

The value of test statistic is 2.50.

Explanation of Solution

Calculation:

In a simple random sample of 500 people who took driving test, only 445 passed the test.

Denote p as the true population proportion who passed the test.

The given test hypotheses are:

Null hypothesis:

 $H_0:p=0.85$.

That is, the true proportion of the people who passed the test is 0.85.

Alternate hypothesis:

$H_1:p>0.85$.

That is, the true proportion of the people who passed the test is greater than 0.85.

Assumptions for performing a hypothesis test for a population proportion:

• The samples taken from the population are simple random samples.
• The population is at least 20 times as large as the sample.
• The samples in the population are divided into two categories.
• The values of $n{\widehat{p}}_0$ and $n{\widehat{q}}_0$ should be greater than or equal to 10.

Requirement check:

• The sample of 500 is simple random samples.
• The information about the population is not known. However, the population size is assumed to be more than 20 times as large as the sample.
• The samples in the population seemed to be categorized into two parts. That is, individuals who passed and individuals who do not passed.
• Verify the condition: $n{\widehat{p}}_0\ge 10$ and $n{\widehat{q}}_0\ge 10$

  Here,

  $\begin{array}{c}{\widehat{p}}_0=\frac{445}{500}\\ =0.89\end{array}$

  Substitute n as 500 and ${\widehat{p}}_0$ as 0.89,

  $\begin{array}{c}n{\widehat{p}}_0=500\left(0.89\right)\\ =445\\ >10\end{array}$

  Substitute n as 500 and ${\widehat{q}}_0$ as $0.11\left(=1-{\widehat{p}}_0\right)$,

  $\begin{array}{c}n{\widehat{q}}_0=500\left(0.11\right)\\ =55\\ >10\end{array}$

Therefore, all the conditions are satisfied.

Test statistic:

The z-test statistic is:

$z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}$,

Where, $\widehat{p}$ be the sample proportion, $p_0$ be the population proportion and n be the sample size.

Software procedure:

Step by step procedure to find the test statistic using the MINITAB software:

• Choose Stat > Basic Statistics > 1-Proportion.
• In Number of trials, enter the Sample size as 445 and in Number of Events, Enter Number of observations as 500.
• Enter Hypothesized proportion as 0.85.
• Select Options.
• Choose $\text{Proportion}>\text{hypothesized proprotion}$ in Alternative hypothesis.
• Click OK in all dialogue boxes.

The output using Minitab is given below:

From the MINITAB output, the test statistic, that is, the z-value is 2.50.

Thus, the value of test statistic z is 2.50.

b.

To determine

Decide whether the null hypothesis $H_0$ is rejected at $\alpha =0.05$ level.

Answer to Problem 8CQ

The null hypothesis $H_0$ is rejected at $\alpha =0.05$ level.

Explanation of Solution

From previous part (a), it has been found that the value of test statistic z is 2.50.

From the given hypothesis, the alternative hypothesis contains the greater than $(>)$ symbol. Thus, it is clear that the hypothesis follows the right-tailed test.

From Table 8.1 “Table of Critical Values”, the critical value for right-tailed test at $\alpha =0.05$ level is 1.645.

For right-tailed test, the critical region will be the observed values which lie above the critical value of 1.645.

Decision based on the critical value method:

For left-tailed test:  If $z\le -z_{\alpha }$, reject $H_0$.

For right-tailed test:  If $z\ge z_{\alpha }$, reject $H_0$.

For two-tailed test:  If $z\le -z_{\frac{\alpha }{2}}\text{or}z\ge z_{\frac{\alpha }{2}}$, reject $H_0$.

Conclusion:

The critical value at $\alpha =0.05$ level is 1.645 and the test statistic is 2.50.

Here, the test statistic value of 2.50 is greater than 1.645, that is, lies in the critical region.

That is, $z\left(=2.50\right)>z_{\alpha }\left(=1.645\right)$.

Therefore, the null hypothesis is rejected.

c.

To determine

State a conclusion.

Answer to Problem 8CQ

The conclusion is that, there is evidence that the true proportion of the people who passed the test is greater than 0.85.

Explanation of Solution

From previous part (b), it has been found that the null hypothesis is rejected.

Hence, there is evidence that the true proportion of the people who passed the test is greater than 0.85.