AP Statistics Test B - Inference for Proportions - Part V Name 1. We have calculated a confidence interval based upon a sample of n=200 Now we want to get a better estimate with a margin of error only one fifth as large. We need a new sample with n at least A) 40 B) 240 C) 450 D) 1000 (E) 5000

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AP Statistics Test B - Inference for Proportions - Part V Name 1. We have calculated a confidence interval based upon a sample of n=200 Now we want to get a better estimate with a margin of error only one fifth as large. We need a new sample with n at least A) 40 B) 240 C) 450 D) 1000 (E) 5000

Answer 1

### E) 5000

Answer 2

## Step 1: Understand Margin of Error ### The margin of error (MOE) is inversely proportional to the square root of the sample size \( n \), specifically \( MOE \propto \frac{1}{\sqrt{n}} \). ## Step 2: Calculate Margin of Error Reduction Factor ### We aim to reduce the margin of error to one fifth of its original size: \( \frac{MOE_{\text{new}}}{MOE_{\text{old}}} = \frac{1}{5} \). ## Step 3: Relate Sample Sizes for New Margin of Error ### Using the relation \( MOE \propto \frac{1}{\sqrt{n}} \), we derive: \[ \frac{1}{\sqrt{n_{\text{new}}}} = \frac{1}{5} \cdot \frac{1}{\sqrt{200}} \] ## Step 4: Solve for the New Sample Size ### Squaring both sides to isolate \( n_{\text{new}} \): \[ \frac{1}{n_{\text{new}}} = \left(\frac{1}{5}\right)^2 \cdot \frac{1}{200} \] \[ \frac{1}{n_{\text{new}}} = \frac{1}{25} \cdot \frac{1}{200} = \frac{1}{5000} \] Thus, \( n_{\text{new}} = 5000 \).

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AP Statistics Test B - Inference for Proportions - Part V Name 1. We have calculated a confidence interval based upon a sample of n=200 Now we want to get a better estimate with a margin of error only one fifth as large. We need a new sample with n at least A) 40 B) 240 C) 450 D) 1000 (E) 5000

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