Question

earch center claims that at least 30% of adults in a certain country think that their taxes will be audited. in a random le of 1100 adults in that country in a recent year, 25% say they are concerned that their taxes will be audited. at .05, is there enough evidence to reject the center’s claim? complete parts (a) through (d) below. a. the rejection region is \\(\square < z < \square\\). b. the rejection regions are \\(z < \square\\) and \\(z > \square\\). c. the rejection region is \\(z < \square\\). d. the rejection region is \\(z > 1.65\\). ) find the standardized test statistic \\(z\\). \\(= -3.62\\) (round to two decimal places as needed.) d) decide whether to reject or fail to reject the null hypothesis and interpret the decision in the context of the original claim. \\(\square\\) the null hypothesis. there \\(\square\\) enough evidence to \\(\square\\) the center’s claim.

Explanation:

Step1: Identify Hypotheses

The claim is \( H_0: p \geq 0.30 \) (null), \( H_a: p < 0.30 \) (alternative, left - tailed test? Wait, no—wait, the rejection region for left - tailed (when \( H_a: p < p_0 \)) at \( \alpha = 0.05 \) has critical value \( z=- 1.645\approx - 1.65 \), but wait, the given option D was \( z > 1.65 \), but let's re - check. Wait, maybe the claim is \( H_0: p \geq 0.30 \), so \( H_a: p < 0.30 \) (left - tailed). But the test statistic is \( z=-3.62 \).

Step2: Rejection Region for Left - Tailed Test

For a left - tailed test with \( \alpha = 0.05 \), the critical value \( z_{\alpha}=-1.645\approx - 1.65 \). The rejection region is \( z < - 1.65 \). But in the options, option C is \( z < \square \). Wait, maybe I misread the hypothesis. Wait, if the claim is "at least 30%", so \( H_0: p\geq0.30 \), \( H_a: p < 0.30 \) (left - tailed). The test statistic \( z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \), where \( \hat{p}=0.25 \), \( p_0 = 0.30 \), \( n = 1100 \).

Calculating \( z=\frac{0.25 - 0.30}{\sqrt{\frac{0.30\times0.70}{1100}}}=\frac{- 0.05}{\sqrt{\frac{0.21}{1100}}}=\frac{-0.05}{\sqrt{0.0001909}}\approx\frac{-0.05}{0.01382}\approx - 3.62 \), which matches the given value.

For a left - tailed test at \( \alpha = 0.05 \), the rejection region is \( z < - 1.645\approx - 1.65 \). But looking at the options, option C is "The rejection region is \( z < \square \)". So we need to fill \( - 1.65 \) (or \( - 1.645 \)). But let's focus on part (d).

Step3: Decision Rule

Compare test statistic \( z=-3.62 \) with critical value. For left - tailed test, critical value \( z_{0.05}=-1.645 \). Since \( - 3.62 < - 1.645 \), we reject the null hypothesis.

Answer:

For part (d): Reject the null hypothesis. There is enough evidence to reject the center’s claim.

So the blanks: "Reject" the null hypothesis. There "is" enough evidence to "reject" the center’s claim.