Question

a research center claims that at least 30% of adults in a certain country think that their taxes will be audited. in a random sample of 1100 adults in that country in a recent year, 25% say they are concerned that their taxes will be audited. at α=0.05, is there enough evidence to reject the center’s claim? complete parts (a) through (d) below. z₀ = | -1.64 | (round to two decimal places as needed. use a comma to separate answers as needed.) identify the rejection region(s). select the correct choice below and fill in the answer box(es) to complete your choice. (round to two decimal places as needed.) a. the rejection region is < z < . b. the rejection regions are z < and z > . c. the rejection region is z < . d. the rejection region is z > . (c) find the standardized test statistic z. z = (round to two decimal places as needed.)

Explanation:

Step1: Identify the hypothesis test type

The claim is "at least 30%", so the null hypothesis \( H_0: p \geq 0.30 \) and the alternative hypothesis \( H_a: p < 0.30 \) (left - tailed test? Wait, no, wait. Wait, the claim is the research center's claim that \( p\geq0.30 \). We are testing if there is enough evidence to reject this claim. So the alternative hypothesis is \( H_a:p < 0.30 \)? Wait, no, wait. Wait, the sample proportion \( \hat{p}=0.25 \), \( n = 1100 \), \( p_0=0.30 \).

The formula for the standardized test statistic \( z \) for a proportion is \( z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \)

Step2: Calculate the numerator and denominator

First, calculate \( \hat{p}-p_0 \): \( \hat{p} = 0.25 \), \( p_0=0.30 \), so \( 0.25 - 0.30=- 0.05 \)

Then, calculate the standard error \( \sqrt{\frac{p_0(1 - p_0)}{n}} \): \( p_0 = 0.30 \), \( 1-p_0 = 0.70 \), \( n = 1100 \)

\( \sqrt{\frac{0.30\times0.70}{1100}}=\sqrt{\frac{0.21}{1100}}=\sqrt{0.0001909}\approx0.0138 \)

Step3: Calculate the z - statistic

Now, \( z=\frac{- 0.05}{0.0138}\approx - 3.62 \) (Wait, let's recalculate the standard error more accurately. \( 0.3\times0.7 = 0.21 \), \( 0.21\div1100\approx0.000190909 \), square root of \( 0.000190909 \) is \( \sqrt{0.000190909}\approx0.013817 \)

Then \( z=\frac{0.25 - 0.30}{0.013817}=\frac{- 0.05}{0.013817}\approx - 3.62 \)

Answer:

\( z\approx - 3.62 \)