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The problem in the image is asking to test the belief that sons are taller than their fathers using a paired data approach. The data contains the heights of 13 fathers and their sons, and the hypothesis test is set at a significance level of \( \alpha = 0.025 \).

The steps to solve this hypothesis test problem are:

1. **Set up hypotheses**:
   - Null hypothesis (\( H_0 \)): \( \mu_d = 0 \) (There is no difference in height between fathers and sons, or the mean difference is zero).
   - Alternative hypothesis (\( H_1 \)): \( \mu_d < 0 \) (Sons are taller than their fathers, meaning the height difference is negative).

2. **Calculate the differences**:
   We need to calculate the differences \( d_i = X_i - Y_i \) for each pair of heights, where \( X_i \) is the father's height and \( Y_i \) is the son's height.

3. **Calculate the mean and standard deviation of the differences**:
   Once we have the differences, compute the sample mean \( \bar{d} \) and the standard deviation \( s_d \).

4. **Test statistic**:
   Use the t-test for paired data to calculate the t-statistic:
   \[
   t = \frac{\bar{d}}{s_d / \sqrt{n}}
   \]
   where \( n \) is the number of pairs (13 in this case).

5. **Find the critical value**:
   Using the t-distribution table at \( \alpha = 0.025 \) with \( n-1 = 12 \) degrees of freedom, find the critical value.

6. **Conclusion**:
   If the computed t-value is less than the critical t-value, we reject the null hypothesis.

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Would you like me to calculate the differences and complete the hypothesis test for you based on this data?

### Related Questions:
1. How do we calculate the standard deviation of the differences?
2. What are the assumptions of a paired t-test?
3. How can we interpret a p-value in this context?
4. Why is it important to check if the data is normally distributed for the t-test?
5. How would we conduct this test using software like Excel or a graphing calculator?

#### Tip:
Always ensure the differences between paired data follow a normal distribution when applying the paired t-test, as this is an underlying assumption for valid results.

To test the belief that sons are taller than their fathers, a student randomly selects 13 fathers who have adult male children. She records the height of both the father and son in inches and obtains the following data. Are sons taller than their fathers? Use the α = 0.025 level of significance. Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

This problem involves testing the belief that sons are taller than their fathers using a paired t-test at a significance level of α = 0.025. The data consists of 13 father-son height pairs. Learn how to calculate the differences, mean, and standard deviation of the height differences, and use the t-statistic to determine the outcome.

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