8. Sampling Distributions & Confidence Intervals: Proportion
Confidence Intervals for Population Proportion
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Your company has asked you to estimate the proportion of people who prefer the color red over other primary colors for manufacturing purposes. If they want the estimate to be within of the true proportion with confidence, how many people should you survey?
A
1825
B
6766
C
9604
D
97
Verified step by step guidance1
Understand the problem: You need to determine the sample size required to estimate the proportion of people who prefer the color red with a specified margin of error and confidence level.
Identify the formula for sample size calculation in proportion estimation: The formula is \( n = \frac{{Z^2 \, p \, (1-p)}}{{E^2}} \), where \( n \) is the sample size, \( Z \) is the Z-score corresponding to the confidence level, \( p \) is the estimated proportion, and \( E \) is the margin of error.
Determine the Z-score for a 95% confidence level: The Z-score for 95% confidence is typically 1.96.
Assume an estimated proportion \( p \): If no prior estimate is available, use \( p = 0.5 \) as it maximizes the sample size.
Substitute the values into the formula: Use \( Z = 1.96 \), \( p = 0.5 \), and \( E = 0.01 \) to calculate the sample size \( n \).
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