To determine the standard deviation (often called the standard error in this context) of the sampling distribution of the difference in sample proportions [tex]\(\hat{p}_A - \hat{p}_T\)[/tex], we proceed as follows:
1. Identify the given proportions:
- Proportion of adults who read nonfiction books: [tex]\( p_A = 0.56 \)[/tex]
- Proportion of teenagers who read nonfiction books: [tex]\( p_T = 0.39 \)[/tex]
2. Identify the sample sizes:
- Number of adults surveyed ([tex]\(n_A\)[/tex]): 28
- Number of teenagers surveyed ([tex]\(n_T\)[/tex]): 41
3. Calculate the standard error of the difference in proportions:
The formula for the standard error (SE) of the difference between two independent sample proportions is:
[tex]\[
SE = \sqrt{\left( \frac{p_A (1 - p_A)}{n_A} \right) + \left( \frac{p_T (1 - p_T)}{n_T} \right)}
\][/tex]
Here we plug in the given values:
[tex]\[
SE = \sqrt{\left( \frac{0.56 \times (1 - 0.56)}{28} \right) + \left( \frac{0.39 \times (1 - 0.39)}{41} \right)}
\][/tex]
4. Simplify the calculations step-by-step:
- Calculate [tex]\(0.56 \times (1 - 0.56) = 0.56 \times 0.44 = 0.2464\)[/tex]
- Calculate [tex]\(0.39 \times (1 - 0.39) = 0.39 \times 0.61 = 0.2379\)[/tex]
- Divide these by their respective sample sizes:
[tex]\[
\frac{0.2464}{28} = 0.0088 \quad \text{and} \quad \frac{0.2379}{41} = 0.0058
\][/tex]
- Sum these results:
[tex]\[
0.0088 + 0.0058 = 0.0146
\][/tex]
- Take the square root of the sum:
[tex]\[
SE = \sqrt{0.0146} \approx 0.121
\][/tex]
5. Interpretation:
The computed standard error of approximately 0.121 means that the difference (adult minus teenager) in the sample proportions of those who read nonfiction books as a hobby will typically vary by about 0.121 from the true difference in proportions.
Therefore, the correct calculation and interpretation is:
- The difference (adult - teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about [tex]\(0.121\)[/tex] from the true difference in proportions.
So, the correct choice is:
The difference (adult - teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.121 from the true difference in proportions.
