A marriage counselor has traditionally seen that the proportion p of all married couples for whom her communication program can prevent divorce is 76%. After making some recent changes, the marriage counselor now claims that her program can prevent divorce in more than 76% of married couples. In a random sample of 205 married couples who completed her program, 171 of them stayed together. Based on this sample, is there enough evidence to support the marriage counselor's claim at the 0.01 level of significance?
Perform a one-tailed test. Then complete the parts below.
Carry vour intermediate computations to three or more decimal places. If necessarv, consult a list of formulas.
Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.)

To determine if there is enough evidence to support the marriage counselor's claim that her program can prevent divorce in more than 76% of married couples, we need to perform a one-tailed hypothesis test for the proportion.

### Step 1: Define the Hypotheses
- **Null Hypothesis (\(H_0\))**: \( p = 0.76 \) (The proportion of couples for whom the program can prevent divorce is 76%.)
- **Alternative Hypothesis (\(H_a\))**: \( p > 0.76 \) (The proportion of couples for whom the program can prevent divorce is greater than 76%.)

### Step 2: Calculate the Sample Proportion
The sample proportion (\(\hat{p}\)) is calculated as:
\[
\hat{p} = \frac{x}{n} = \frac{171}{205} \approx 0.8341
\]

### Step 3: Determine the Test Statistic
The test statistic for a proportion is given by:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
Where:
- \( p_0 = 0.76 \) (the hypothesized population proportion)
- \( \hat{p} = 0.8341 \) (the sample proportion)
- \( n = 205 \) (the sample size)

First, calculate the standard error (SE):
\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.76(1 - 0.76)}{205}} \approx 0.0297
\]

Now, compute the z-score:
\[
z = \frac{0.8341 - 0.76}{0.0297} \approx 2.4933
\]

### Step 4: Find the Critical Value
At the 0.01 significance level for a one-tailed test, the critical z-value (z*) is:
- \( z^* \) for \( \alpha = 0.01 \) (one-tailed) ≈ **2.33**

### Step 5: Make a Decision
- If the calculated \( z \)-value is greater than the critical \( z^* \)-value, we reject the null hypothesis.

Since \( z = 2.4933 \) is greater than \( z^* = 2.33 \), we reject the null hypothesis.

### Step 6: Conclusion
There is enough evidence at the 0.01 significance level to support the marriage counselor's claim that her program can prevent divorce in more than 76% of married couples.

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Do you have any questions, or would you like more details? 

Here are five related questions:
1. What is the importance of the significance level in hypothesis testing?
2. How would the conclusion change if the significance level was 0.05 instead of 0.01?
3. What is the meaning of a one-tailed test compared to a two-tailed test?
4. How do sample size and population proportion affect the z-score?
5. What are the potential limitations of using hypothesis testing in real-world scenarios?

**Tip:** In hypothesis testing, always check the assumptions behind the test, such as the sample size and randomness, to ensure the test's validity.

Explore a one-tailed hypothesis test to determine if a marriage counselor's program can prevent divorce in more than 76% of couples. Includes calculations of sample proportion, test statistic, critical value, and conclusion at the 0.01 significance level.

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