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In a nearby town, [tex]$56\%$[/tex] of adults read nonfiction books as a hobby, while only [tex]$39\%$[/tex] of teenagers read nonfiction books as a hobby. Let [tex]$\hat{p}_A$[/tex] and [tex]$\hat{p}_T$[/tex] be the sample proportions of adults and teenagers, respectively, who read nonfiction books as a hobby. Suppose 28 adults and 41 teenagers from this town are selected at random and asked if they read nonfiction books as a hobby.

Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of [tex]$\hat{p}_A - \hat{p}_T$[/tex]?

A. The difference (adult - teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.003 from the true difference in proportions.

B. The difference (adult - teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.015 from the true difference in proportions.

C. The difference (adult - teenager) in the sample proportions of those who read nonfiction books as a hobby typically varies about 0.055 from the true difference in proportions.

D. 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.



Answer :

GinnyAnswer
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.

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