wishes the estimate to be within 0.04 with 95% confidence if (a) she uses a previous estimate of 0.34? (b) she does not use any prior estimates? size sample should be obtained if she
wishes the estimate to be within 0.04 with 95% confidence if (a) she uses a previous estimate of 0.34? (b) she does not use any prior estimates? size sample should be obtained if she
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
I need help with A and B.
![**Title: Estimating Sample Size for Research on High-Speed Internet Access**
**Objective:**
To determine the required sample size for estimating the proportion of adults with high-speed internet access, ensuring the estimate is within a margin of error of 0.04 with 95% confidence.
**Scenario:**
A researcher aims to estimate the proportion of adults with high-speed internet access. She wants the estimate to be accurate within ±0.04 with 95% confidence. The sample size needed depends on two different conditions:
**Conditions:**
(a) If a previous estimate of 0.34 is used.
(b) Without any prior estimates.
**Steps for Calculation:**
1. **With a Previous Estimate:**
- Given:
- Previous estimate (\( \hat{p} \)) = 0.34
- Margin of error (E) = 0.04
- Confidence level = 95% (Z-value corresponding to 95% confidence level = 1.96)
2. **Without Any Prior Estimate:**
- In absence of prior estimate, the proportion (\( \hat{p} \)) is taken as 0.5 for maximum variability.
**Formula for Sample Size:**
\[ n = \left( \frac{Z^2 \cdot \hat{p} \cdot (1 - \hat{p})}{E^2} \right) \]
Where:
- \( n \): Sample size
- \( Z \): Z-value for the given confidence level (1.96 for 95% confidence interval)
- \( \hat{p} \): Estimated proportion
- \( E \): Margin of error
**Interactive Component:**
- Click the icon to view the table of critical values.
**Solution for Part (a):**
\[ n = \left( \frac{(1.96)^2 \cdot 0.34 \cdot (1 - 0.34)}{0.04^2} \right) \]
(Round up to the nearest integer for final sample size)
**Solution for Part (b):**
\[ n = \left( \frac{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}{0.04^2} \right) \]
(Round up to the nearest integer for final sample size)
**Note:**
It's crucial](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F254f01d1-5636-4aee-8bdd-3770d49eb7a5%2F81536891-23ec-47d0-b5b3-cdbb28ed16f2%2F9htvdbj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Estimating Sample Size for Research on High-Speed Internet Access**
**Objective:**
To determine the required sample size for estimating the proportion of adults with high-speed internet access, ensuring the estimate is within a margin of error of 0.04 with 95% confidence.
**Scenario:**
A researcher aims to estimate the proportion of adults with high-speed internet access. She wants the estimate to be accurate within ±0.04 with 95% confidence. The sample size needed depends on two different conditions:
**Conditions:**
(a) If a previous estimate of 0.34 is used.
(b) Without any prior estimates.
**Steps for Calculation:**
1. **With a Previous Estimate:**
- Given:
- Previous estimate (\( \hat{p} \)) = 0.34
- Margin of error (E) = 0.04
- Confidence level = 95% (Z-value corresponding to 95% confidence level = 1.96)
2. **Without Any Prior Estimate:**
- In absence of prior estimate, the proportion (\( \hat{p} \)) is taken as 0.5 for maximum variability.
**Formula for Sample Size:**
\[ n = \left( \frac{Z^2 \cdot \hat{p} \cdot (1 - \hat{p})}{E^2} \right) \]
Where:
- \( n \): Sample size
- \( Z \): Z-value for the given confidence level (1.96 for 95% confidence interval)
- \( \hat{p} \): Estimated proportion
- \( E \): Margin of error
**Interactive Component:**
- Click the icon to view the table of critical values.
**Solution for Part (a):**
\[ n = \left( \frac{(1.96)^2 \cdot 0.34 \cdot (1 - 0.34)}{0.04^2} \right) \]
(Round up to the nearest integer for final sample size)
**Solution for Part (b):**
\[ n = \left( \frac{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}{0.04^2} \right) \]
(Round up to the nearest integer for final sample size)
**Note:**
It's crucial
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