For the following hypothesis​ test:

Start 3 By 1 Matrix 1st Row 1st Column Upper H 0 : p equals 0.40 2nd Row 1st Column Upper H Subscript Upper A Baseline : p not equals 0.40 3rd Row 1st Column alpha equals 0.01 EndMatrix

H0: p=0.40

HA: p≠0.40

α=0.01

With

nequals=64

and

pequals=​0.42,

state the decision rule in terms of the critical value of the test statistic.

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Part 1

A.

The decision rule​ is: reject the null hypothesis if the calculated value of the test​ statistic, z, is greater than 2.575 or less than

minus−2.575.

​Otherwise, do not reject.

B.

The decision rule​ is: reject the null hypothesis if the calculated value of the test​ statistic, z, is less than 2.575 or greater than

minus−2.575.

​Otherwise, do not reject.

C.

The decision rule​ is: reject the null hypothesis if the calculated value of the test​ statistic, z, is less than 2.013 or greater than

minus−2.013.

​Otherwise, do not reject.

D.

The decision rule​ is: reject the null hypothesis if the calculated value of the test​ statistic, z, is greater than 2.013 or less than

minus−2.013.

​Otherwise, do not reject.

To determine the decision rule for this hypothesis test, let's go step by step:

### 1. **Null and Alternative Hypotheses**:
- \( H_0: p = 0.40 \)
- \( H_A: p \neq 0.40 \)

This is a two-tailed hypothesis test because the alternative hypothesis states that \( p \) is not equal to 0.40.

### 2. **Significance Level (α)**:
- \( \alpha = 0.01 \)

This means we have a 1% significance level, and since it's a two-tailed test, we will split the significance level equally between both tails (0.005 in each tail).

### 3. **Critical Values for a Two-Tailed Test**:
For a two-tailed test at \( \alpha = 0.01 \), we need to find the critical z-values that correspond to the lower and upper tails.

- Using a standard normal distribution table, the z-value that corresponds to the cumulative probability of \( 0.005 \) in the lower tail is approximately **−2.575**, and the z-value for \( 0.995 \) (upper tail) is **+2.575**.

### 4. **Decision Rule**:
- We reject the null hypothesis \( H_0 \) if the calculated test statistic \( z \) is **less than −2.575 or greater than +2.575**.
- Otherwise, we do not reject \( H_0 \).

### 5. **Correct Answer**:
Thus, the correct decision rule is:

**A**. The decision rule is: reject the null hypothesis if the calculated value of the test statistic \( z \) is greater than 2.575 or less than −2.575. Otherwise, do not reject.

Would you like a detailed explanation of how to compute the z-test statistic for this test? 

Here are five related questions to help deepen your understanding:

1. How do we calculate the z-test statistic in a hypothesis test for proportions?
2. Why do we split the significance level in a two-tailed test?
3. What is the importance of using the standard normal distribution in hypothesis testing?
4. How would the decision rule change if the significance level \( \alpha \) were 0.05?
5. What are the implications of rejecting \( H_0 \) in this context?

**Tip:** Always check whether the test is one-tailed or two-tailed before determining the critical values!

For the following hypothesis​ test:

H0: p = 0.40
HA: p ≠ 0.40
α = 0.01
n = 64 and p = 0.42, state the decision rule in terms of the critical value of the test statistic.

A. The decision rule is: reject the null hypothesis if the calculated value of the test statistic, z, is greater than 2.575 or less than −2.575. Otherwise, do not reject.

B. The decision rule is: reject the null hypothesis if the calculated value of the test statistic, z, is less than 2.575 or greater than −2.575. Otherwise, do not reject.

C. The decision rule is: reject the null hypothesis if the calculated value of the test statistic, z, is less than 2.013 or greater than −2.013. Otherwise, do not reject.

D. The decision rule is: reject the null hypothesis if the calculated value of the test statistic, z, is greater than 2.013 or less than −2.013. Otherwise, do not reject.

Learn how to perform a two-tailed z-test for a proportion with H0: p = 0.40, HA: p ≠ 0.40, significance level α = 0.01, sample size n = 64, and sample proportion p̂ = 0.42. Understand the decision rule based on critical z-values ±2.575.

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