The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one observed under the null hypothesis. For a one-tailed z-test, we calculate the p-value based on the cumulative distribution function (CDF) of the standard normal distribution.

Given that we have a positive z-value, the p-value is:

\[
\text{p-value} = 1 - \Phi(z)
\]

Where \(\Phi(z)\) is the cumulative probability up to the z-value.

Let's calculate the p-value.from scipy.stats import norm

# Calculate the p-value for the one-tailed test
p_value = 1 - norm.cdf(z)
p_valueThe p-value is approximately \(0.00695\).

### Conclusion:
Since the p-value (\(0.00695\)) is much smaller than the significance level (\(\alpha = 0.10\)), we have strong evidence to reject the null hypothesis. This further supports the marriage counselor's claim that her program can prevent divorce in more than 75% of married couples.

Would you like further details or explanations?

### Related Questions
1. How is the p-value interpreted in hypothesis testing?
2. What is the relationship between the p-value and the significance level?
3. How does the p-value change with different sample sizes?
4. Why is the p-value different for one-tailed vs. two-tailed tests?
5. What are Type I and Type II errors in hypothesis testing?
6. How would the results differ with a significance level of 0.05?
7. What is the difference between statistical significance and practical significance?
8. How do confidence intervals relate to hypothesis testing?

**Tip:** Always compare the p-value with the chosen significance level (\(\alpha\)) to make a decision on the hypothesis test.

Learn how to calculate and interpret p-values in hypothesis testing using the normal distribution and one-tailed tests. Includes a detailed explanation and example calculation with a p-value of approximately 0.00695, demonstrating strong evidence against the null hypothesis.

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