Question

Testing Claims About Proportions. In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Bias in Jury SelectionIn the case of Casteneda v. Partida,it was found that during a period of 11 years in Hidalgo County, Texas, 870 people were selected for grand jury duty and 39% of them were Americans of Mexican ancestry. Among the people eligible for grand jury duty, 79.1% were Americans of Mexican ancestry. Use a 0.01 significance level to test the claim that the selection process is biased against Americans of Mexican ancestry. Does the jury selection system appear to be biased?

Short answer

The hypotheses are as follows.

$H_0:p=0.791$(Null hypothesis)

$H_1:p<0.791$(Alternative hypothesis and original claim)

The test statistic is -29.09. The critical value is -2.33, and the p-value is 0.0001.

There is enough evidence to prove that the selection process is biased against the Americans of Mexican ancestry. Hence, the jury selection system appears to be biased.

Step-by-step solution

Given information

A sample survey of 870 people is recorded. Therefore, sample size $n=870$.

The sample proportion of Americans with Mexican ancestry is $\hat{p}=0.39$.

Among all the people in the grand jury duty, the proportion of Americans with Mexican ancestry is $0.791$.

The level of significance $\alpha =0.01$

Check the requirements

The requirements for testing the claim for proportions are stated below.

1. Assume that 870 adults are selected randomly.
2. They are fixed of independent trails.
3. The requirements $np⩾5$and $nq⩾5$are satisfied as follows.

$$\begin{gathered} np=870\times 0.791 \\ =688.17 \\ ⩾5 \end{gathered}$$

$$\begin{gathered} nq=870\times (1-0.791) \\ =181.83 \\ ⩾5 \end{gathered}$$

So, the three requirements for the z-test for proportions are satisfied.

State the hypotheses

Let p be the actual proportion of the selected people who are Americans of Mexican ancestry.

The hypotheses are as follows.

$H_0:p=0.791$(Null hypothesis)

$H_1:p<0.791$(Alternative hypothesis and original claim)

As the original claim contains a less-than symbol,it is a left-tailed test.

Compute the test statistic

The sampling distribution of sample proportions can be approximated by the normal distribution.

The test statistic is given as follows.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.39-0.791}{\sqrt{\frac{0.791\times (1-0.791)}{870}}} \\ =-29.0899 \end{gathered}$$

The test statistic value is -29.09.

Compute the critical value and the p-value

This is a left-tailed test. So, the area of the critical region is an area of in the left tail. Referring to Table A-2 (standard normal table),the z-score value corresponding to 0.0100 corresponds to rows -2.3 and 0.03, which implies that its value is -2.33.

Thus, the critical value is -2.33.

Also, for the left-tailed test, the p-value is computed as 0.0001, corresponding to rows -3.5 and above.

Thus,

$$\begin{gathered} P-value=P\left(Z<-29.09\right) \\ =0.0001 \end{gathered}$$

Decision rule

The decision rule states the following:

If the test statistic value is lesser than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

In terms of p-value, reject the null hypothesis if the p-value is lesser than the level of significance. Otherwise, fail to reject the null hypothesis.

In this scenario, 0.0001 is lesser than 0.01. Hence, reject the null hypothesis.

State the conclusion

As we reject the null hypothesis, there is sufficient evidence to support the claim that the population proportion is lesser than 79.1%.

Therefore, the process is biased against the Americans of Mexican ancestry, and the jury selection system appears to be biased.