Questions: Katrina wants to estimate the proportion of adults who read at least 10 books last year. To do so, she obtains a simple random sample of 100 adults and constructs a 95% confidence interval. Matthew also wants to estimate the proportion of adults who read at least 10 books last year. He obtains a simple random sample of 400 adults and constructs a 99% confidence interval. Assuming both Katrina and Matthew obtained the same point estimate, whose estimate will have the smaller margin of error? Justify your answer. Whose estimate will have the smaller margin of error and why? A. Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence. B. Matthew's estimate will have the smaller margin of error because the sample size is larger and the level of confidence is higher. C. Katrina's estimate will have the smaller margin of error because the lower level of confidence more than compensates for the smaller sample size. D. Katrina's estimate will have the smaller margin of error because the sample size is smaller and the level of confidence is lower.

Katrina wants to estimate the proportion of adults who read at least 10 books last year. To do so, she obtains a simple random sample of 100 adults and constructs a 95% confidence interval. Matthew also wants to estimate the proportion of adults who read at least 10 books last year. He obtains a simple random sample of 400 adults and constructs a 99% confidence interval. Assuming both Katrina and Matthew obtained the same point estimate, whose estimate will have the smaller margin of error? Justify your answer. Whose estimate will have the smaller margin of error and why? A. Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence. B. Matthew's estimate will have the smaller margin of error because the sample size is larger and the level of confidence is higher. C. Katrina's estimate will have the smaller margin of error because the lower level of confidence more than compensates for the smaller sample size. D. Katrina's estimate will have the smaller margin of error because the sample size is smaller and the level of confidence is lower.
Transcript text: Katrina wants to estimate the proportion of adults who read at least 10 books last year. To do so, she obtains a simple random sample of 100 adults and constructs a $95 \%$ confidence interval. Matthew also wants to estimate the proportion of adults who read at least 10 books last year. He obtains a simple random sample of 400 adults and constructs a $99 \%$ confidence interval. Assuming both Katrina and Matthew obtained the same point estimate, whose estimate will have the smaller margin of error? Justify your answer. Whose estimate will have the smaller margin of error and why? A. Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence. B. Matthew's estimate will have the smaller margin of error because the sample size is larger and the level of confidence is higher. C. Katrina's estimate will have the smaller margin of error because the lower level of confidence more than compensates for the smaller sample size. D. Katrina's estimate will have the smaller margin of error because the sample size is smaller and the level of confidence is lower.
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Solution

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Solution Steps

Step 1: Calculate the Margin of Error for Katrina

For Katrina's estimate, we use the formula for the margin of error:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} \]

Where:

  • \( Z = 1.96 \) (Z-score for a \( 95\% \) confidence level)
  • \( \sigma = 0.5 \) (assumed standard deviation)
  • \( n = 100 \) (sample size)

Substituting the values:

\[ \text{Margin of Error} = \frac{1.96 \times 0.5}{\sqrt{100}} = \frac{0.98}{10} = 0.098 \]

Thus, Katrina's margin of error is \( 0.098 \).

Step 2: Calculate the Margin of Error for Matthew

For Matthew's estimate, we again use the margin of error formula:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} \]

Where:

  • \( Z = 2.5758 \) (Z-score for a \( 99\% \) confidence level)
  • \( \sigma = 0.5 \) (assumed standard deviation)
  • \( n = 400 \) (sample size)

Substituting the values:

\[ \text{Margin of Error} = \frac{2.5758 \times 0.5}{\sqrt{400}} = \frac{1.2879}{20} = 0.0644 \]

Thus, Matthew's margin of error is \( 0.0644 \).

Step 3: Compare the Margins of Error

Now we compare the two margins of error:

  • Katrina's Margin of Error: \( 0.098 \)
  • Matthew's Margin of Error: \( 0.0644 \)

Since \( 0.0644 < 0.098 \), Matthew's estimate has the smaller margin of error.

Final Answer

The answer is \( \boxed{\text{Matthew's estimate will have the smaller margin of error.}} \)

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