Chapter 10, Problem 68CR

To determine

Check whether there is convincing evidence to support the claim that fewer than half of adult in Country A can name at least one justice currently serving on the Supreme Court.

Answer to Problem 68CR

The conclusion is that there is convincing evidence to support the claim that fewer than half of adult in Country A can name at least one justice currently serving on the Supreme Court.

Explanation of Solution

Calculation:

In the survey, out of 1,000 adults in Country A, 410 can name at least one justice currently serving on the Supreme Court.

Step 1:

Population characteristic of interest:

Let p be the population characteristic of interest.

In this context, p denotes population proportion of adults in Country A who can name at least one justice currently serving on the Supreme Court.

Step 2:

Null hypothesis:

$H_0:p=0.5$

That is, there is no convincing evidence to support the claim that fewer than half of adult in Country A can name at least one justice currently serving on the Supreme Court.

Step 3:

Alternative hypothesis:

$H_a:p<0.5$

That is, there is convincing evidence to support the claim that fewer than half of adult in Country A can name at least one justice currently serving on the Supreme Court.

Step 4:

Significance level, $\alpha$:

It is given that the significance level, $\alpha =0.01$.

Step 5:

Test statistic, z:

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

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

Substitute the hypothesized proportion, $p=0.5$ in the test statistic,

$z=\frac{\widehat{p}-0.5}{\sqrt{\frac{0.5\left(1-0.5\right)}{n}}}$

Here, the sample proportion $\widehat{p}$ is not known.

Step 6:

Assumptions:

• Let $\widehat{p}$ be the sample proportion, which is selected from a random sample.
• The large sample z test can be used if the sample size n satisfies the below conditions: $n\left(\text{hypothesized value}\right)\ge 10$ and $n\left(1-\text{hypothesized value}\right)\ge 10$.
• The sample size should not be greater than 10% of the population size.

Requirement check:

• It is assumed that the sample is from a random sample.
• Check the conditions: $n\left(\text{hypothesized value}\right)\ge 10$ and $n\left(1-\text{hypothesized value}\right)\ge 10$.

$\begin{array}{c}n\left(\text{hypothesized value}\right)=np\\ =1,000\left(0.50\right)\\ =\text{500}\\ >10\end{array}$

$\begin{array}{c}n\left(1-\text{hypothesized value}\right)=n\left(1-p\right)\\ =1,000\left(1-0.50\right)\\ =1,000\left(0.50\right)\\ =\text{500}\\ >10\end{array}$

Since $n\left(\text{hypothesized value}\right)$ and $n\left(1-\text{hypothesized value}\right)$ are greater than 10, the sample size requirement is met.

Therefore, the large sample z test is appropriate.

• Although, the population size is not known, it is reasonable to assume that the sample size of 1,000 adults in Country A as the representative for all adults in Country A. It is also definite that the sample size is less than the 10% of the population of the adults in Country A.

Step 7:

Software procedure:

Step-by-step procedure to obtain the test statistic using MINITAB:

• Choose Stat > Basic Statistics > 1 Proportion.
• Choose Summarized data.
• In Number of events, enter 430. In Number of trials, enter 1,000.
• Check Perform hypothesis test. In Hypothesized proportion, enter 0.5.
• Click Options. Under Alternative, and choose less than.
• Click OK in each dialog box.

Output using MINITAB software is given below:

From the given MINITAB output, the value of test statistic is –4.43.

Step 8:

P-value:

From the MINITAB output, the P-value is 0.

Step 9:

Decision rule:

If $P\text{-value}\le \alpha$, then reject the null hypothesis $H_0$.

Here, the $P\text{-value}$ of 0 is less than the significance level 0.01.

That is, $P\text{-value}\left(=0\right)<\alpha \left(=0.01\right)$.

The decision is that the null hypothesis is rejected.

Conclusion:

Hence, there is convincing evidence to support the claim that fewer than half of adult in Country A can name at least one justice currently serving on the Supreme Court.