A marriage counselor has traditionally seen that the proportion p of all married couples for whom her communication program can prevent a divorce is 75%. After making some recent changes, the marriage counselor now claims that her program can prevent divorce in more than 75% of married couples. In a random sample of 205 married couples who completed her program, 169 of them stayed together. Based on this sample, is there enough evidence to support the marriage counselors claim at the 0.10 level of significance? Perform a one-tailed test. Then Complete the Parts below. Carry your immediate computations to three or more decimal places

To determine if there is enough evidence to support the marriage counselor's claim at the 0.10 level of significance, we will perform a hypothesis test for proportions. The hypothesis test will be a one-tailed test since the counselor claims that her program prevents divorce in *more than* 75% of couples. Here are the steps:

### Step 1: Define the hypotheses

- **Null Hypothesis (\(H_0\)):** The proportion of couples for whom the program prevents divorce is still 75%. So, \(H_0: p = 0.75\).
- **Alternative Hypothesis (\(H_a\)):** The proportion of couples for whom the program prevents divorce is more than 75%. So, \(H_a: p > 0.75\).

### Step 2: Determine the test statistic

We will use the **z-test for proportions** because we are dealing with proportions from a large sample.

The formula for the z-test statistic for a proportion is:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]

Where:
- \(\hat{p}\) is the sample proportion.
- \(p_0\) is the hypothesized population proportion (0.75 in this case).
- \(n\) is the sample size.

### Step 3: Compute the sample proportion

The sample proportion (\(\hat{p}\)) is calculated by dividing the number of couples who stayed together by the total number of couples in the sample:

\[
\hat{p} = \frac{169}{205} \approx 0.824
\]

### Step 4: Compute the test statistic

We now substitute the values into the z-test formula:

- \(p_0 = 0.75\)
- \(\hat{p} = 0.824\)
- \(n = 205\)

Let's compute the test statistic \(z\).# Given values
p_hat = 169 / 205  # Sample proportion
p_0 = 0.75  # Null hypothesis proportion
n = 205  # Sample size

# Compute the test statistic (z)
z = (p_hat - p_0) / ((p_0 * (1 - p_0) / n) ** 0.5)
zThe computed test statistic \(z\) is approximately \(2.460\).

### Step 5: Determine the critical value and p-value

For a one-tailed test at the 0.10 significance level, we look for the critical value from the standard normal distribution. The critical value \(z_{\text{crit}}\) for a 0.10 significance level (in a one-tailed test) is approximately \(1.282\).

### Step 6: Compare the test statistic to the critical value

- If \(z > z_{\text{crit}}\), we reject the null hypothesis.
- If \(z \leq z_{\text{crit}}\), we fail to reject the null hypothesis.

Since \(z = 2.460\) is greater than the critical value \(1.282\), we reject the null hypothesis.

### Step 7: Conclusion

There is enough evidence at the 0.10 level of significance to support the marriage counselor's claim that her program can prevent divorce in more than 75% of married couples.

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Would you like more details, or do you have any questions?

### Related Questions
1. What would the result be at a 0.05 significance level?
2. How do we interpret a p-value in hypothesis testing?
3. How does sample size affect the outcome of hypothesis tests?
4. What are the assumptions for the z-test for proportions?
5. How would a two-tailed test differ from a one-tailed test in this scenario?
6. What happens if the null hypothesis is not rejected?
7. What role does the significance level play in hypothesis testing?
8. How does the z-test compare to other tests like the t-test?

**Tip:** Always double-check if your chosen significance level aligns with the importance of avoiding a false conclusion (Type I error).

Understand how to perform a one-tailed hypothesis test for proportions using the Z-test. Explore a real-world example where a marriage counselor claims to prevent divorce in more than 75% of couples. Learn step-by-step calculations and interpretation.

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