A research center claims that at least $$ 29 \% $$ of adults in a certain country think that their taxes will be audited. In a random sample of 1100 adults in th country in a recent year, $$ 24 \% $$ say they are concerned that thoir taxes will be audited. At $$ \alpha=0.10 $$, is there enough evidence to reject the center's clai Complete parts (a) through (d) below. 2. Ine percentage or aauts in the country who think that their taxes will De auarea is not $$ \square $$ \%. C. At least $$ 20 \% $$ of adulte in the coumtry think that their taxes will be audited. D. $$ \% $$ of adults in the country think that their taxes will be audited. Let p be the population proportion of successes, where a success is an adult in the country who thinks that their taxes will be audited. State $$ H_{0} $$ an Select the correct choice below and fill in the answer boxes to complete your choice. (Round to two decimal places as needed.) A. $$ H_{0}: p $$ $$ \square $$ E E. $$ H_{0}: p \geq 0.29 $$ $$ H_{a}: p \leq $$ $$ \square $$ C. $$ H_{0}: p \leq $$ $$ \square $$ $$ H_{a}: p $$ $$ \square $$ F. $$ H_{0}: p= $$ $$ \square $$ $$ \mathrm{H}_{a}: \mathrm{p} \neq $$ (b) Find the critical value(s) and identify the rejection region(s). Identify the critical value(s) for this test. $$ z_{0}=\square $$ (Round to two decimal places as needed. Use a comma to separate answers as needed.)

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A research center claims that at least $$ 29 \% $$ of adults in a certain country think that their taxes will be audited. In a random sample of 1100 adults in th country in a recent year, $$ 24 \% $$ say they are concerned that thoir taxes will be audited. At $$ \alpha=0.10 $$, is there enough evidence to reject the center's clai Complete parts (a) through (d) below. 2. Ine percentage or aauts in the country who think that their taxes will De auarea is not $$ \square $$ \%. C. At least $$ 20 \% $$ of adulte in the coumtry think that their taxes will be audited. D. $$ \% $$ of adults in the country think that their taxes will be audited. Let p be the population proportion of successes, where a success is an adult in the country who thinks that their taxes will be audited. State $$ H_{0} $$ an Select the correct choice below and fill in the answer boxes to complete your choice. (Round to two decimal places as needed.) A. $$ H_{0}: p< $$ $$ \square $$ B. $$ H_{0}: p \neq $$ $$ \square $$ $$ H_{a}: p \geq $$ $$ \square $$ $$ H_{3}: p= $$ $$ \square $$ D. $$ H_{0}: p> $$ $$ \square $$ E E. $$ H_{0}: p \geq 0.29 $$ $$ H_{a}: p \leq $$ $$ \square $$ C. $$ H_{0}: p \leq $$ $$ \square $$ $$ H_{a}: p> $$ $$ \square $$ F. $$ H_{0}: p= $$ $$ \square $$ $$ \mathrm{H}_{a}: \mathrm{p} \neq $$ (b) Find the critical value(s) and identify the rejection region(s). Identify the critical value(s) for this test. $$ z_{0}=\square $$ (Round to two decimal places as needed. Use a comma to separate answers as needed.)

UpStudy AI · Accepted Answer

Let's address the hypothesis testing problem step by step.

### **Context Recap**
- **Claim by Research Center:** At least $$ 29\% $$ of adults in a certain country believe their taxes will be audited.
- **Sample Data:** In a random sample of **1100** adults, $$ 24\% $$ expressed concern about their taxes being audited.
- **Significance Level:** $$ \alpha = 0.10 $$

### **Part (a): State the Null and Alternative Hypotheses**

**Objective:** Determine whether there's enough evidence to reject the research center's claim.

**Definitions:**
- **Population Proportion ($$ p $$)**: The true proportion of adults who believe their taxes will be audited.
- **Null Hypothesis ($$ H_0 $$)**: Represents the research center's claim.
- **Alternative Hypothesis ($$ H_a $$)**: Represents the opposing claim based on sample evidence.

**Given Choices:**

- **E.**
  - $$ H_{0}: p \geq 0.29 $$
  - $$ H_{a}: p \leq 0.29 $$

**Selection and Justification:**

The research center claims that **at least** $$ 29\% $$ believe their taxes will be audited. This translates to:

- **Null Hypothesis ($$ H_0 $$)**: $$ p \geq 0.29 $$ (supports the center's claim)
- **Alternative Hypothesis ($$ H_a $$)**: $$ p < 0.29 $$ (suggests the true proportion is less)

**Note:** Choice E aligns closely, though it presents $$ H_a $$ as $$ p \leq 0.29 $$, which is acceptable for hypothesis testing purposes.

**Final Hypotheses:**

- **$$ H_{0}: p \geq 0.29 $$**
- **$$ H_{a}: p < 0.29 $$**

### **Part (b): Determine the Critical Value(s) and Rejection Region**

**Objective:** Find the z-score that marks the boundary of the rejection region for the null hypothesis at $$ \alpha = 0.10 $$.

**Test Type:** 
- Since we're testing if the proportion is **less than** $$ 29\% $$, this is a **left-tailed** test.

**Critical Value Calculation:**

For a left-tailed test at $$ \alpha = 0.10 $$:

1. **Find the z-score that corresponds to the cumulative probability of $$ \alpha $$.**

$$
   P(Z < z) = 0.10
   $$

2. **Using the standard normal distribution table or a calculator:**

$$
   z_{0.10} \approx -1.28
   $$

**Rejection Region:**

- **Reject $$ H_0 $$** if $$ z $$ **is less than** $$ -1.28 $$.

**Final Critical Value:**

$$
z_{0} = -1.28
$$

**Summary:**

- **Null Hypothesis ($$ H_0 $$)**: $$ p \geq 0.29 $$
- **Alternative Hypothesis ($$ H_a $$)**: $$ p < 0.29 $$
- **Critical Value ($$ z_{0} $$)**: $$ -1.28 $$
- **Rejection Region:** $$ z < -1.28 $$
**Hypotheses:** - **Null Hypothesis ($$ H_0 $$)**: $$ p \geq 0.29 $$ - **Alternative Hypothesis ($$ H_a $$)**: $$ p < 0.29 $$ **Critical Value:** $$ z_{0} = -1.28 $$ **Rejection Region:** $$ z < -1.28 $$

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