Chapter 8.4, Problem 9E

a.

To determine

Find the sample proportion $\widehat{p}$.

Answer to Problem 9E

The value of sample proportion $\widehat{p}$ is 0.675.

Explanation of Solution

In a simple random sample of 80 individuals, only 54 individuals are in the group of interest.

The sample proportion $\widehat{p}$ is:

$\widehat{p}=\frac{x}{n}$

Where, x be the number of observations, n be the sample size.

Substitute the corresponding values to get the proportion,

$\begin{array}{c}\widehat{p}=\frac{54}{80}\\ =0.675\end{array}$

Thus, the sample proportion $\widehat{p}$ is 0.675.

b.

To determine

Check whether the assumptions for the hypothesis test are satisfied.

Answer to Problem 9E

Yes, the assumptions for the hypothesis test are satisfied.

Explanation of Solution

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 80 is obtained by simple random sampling.
• The information about the population size is not known. 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 under the category of interest and individuals not under the category of interest.
• Verify the condition: $n{\widehat{p}}_0\ge 10$ and $n{\widehat{q}}_0\ge 10$

  Substitute n as 80 and ${\widehat{p}}_0$ as 0.675,

  $\begin{array}{c}n{\widehat{p}}_0=80\left(0.675\right)\\ =54\\ >10\end{array}$

  Substitute n as 80 and ${\widehat{q}}_0$ as $0.325\left(=1-{\widehat{p}}_0\right)$,

  $\begin{array}{c}n{\widehat{q}}_0=80\left(0.325\right)\\ =26\\ >10\end{array}$

Therefore, all the conditions are satisfied.

c.

To determine

Find the value of test statistic.

Answer to Problem 9E

The value of test statistic is –2.8.

Explanation of Solution

Calculation:

Denote p as the true population proportion.

The given test hypotheses are:

Null hypothesis:

 $H_0:p=0.8$.

That is, the true proportion of the individuals under the category of interest is 0.8.

Alternate hypothesis:

$H_1:p<0.8$.

That is, the true proportion of the individuals under the category of interest is less than 0.8.

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 80 and in Number of Events, Enter Number of observations as 54.
• Enter Hypothesized proportion as 0.8.
• 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 –2.8.

Thus, the value of test statistic z is –2.8.

d.

To determine

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

Answer to Problem 9E

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

Explanation of Solution

From previous part (c), it has been found that the value of test statistic z is –2.80.

From the given hypothesis, the alternative hypothesis contains the less than $(<)$ symbol. Thus, it is clear that the hypothesis conducts the left-tailed test.

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

For left-tailed test, the critical region will be the observed values which lie below the critical value of –1.645.

Therefore, the critical region is $z\le -1.645$.

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 –2.80.

Here, the test statistic value of –2.80 is less than –1.645, that is, lies in the critical region.

That is, $z\left(=-2.80\right)<z_{\alpha }\left(=-1.645\right)$.

Therefore, the null hypothesis is rejected.

Hence, there is evidence that the true proportion of the individuals under the category of interest is less than 0.80.