Find the sample size needed to estimate the percentage of...

Name: Problem 29, Chapter 7: Elementary Statistics
Uploaded: 2021-08-28T17:47:55-08:00
Duration: 3 min 51 s
Channel: Willis James
Description: Find the sample size needed to estimate the percentage of Republicans among registered voters in California. Use a 0.03 margin of error, use a confidence level of 90%, and assume that p ^ and q ^ are unknown.

Find the sample size needed to estimate the percentage of Republicans among registered voters in California. Use a 0.03 margin of error, use a confidence level of 90%, and assume that p ^ and q ^ are unknown.

Key Concepts

Critical Value (Z-score)

The critical value, or z-score, corresponds to the desired confidence level and is used in the sample size formula to account for the distribution of the sample proportion. For a 90% confidence level, the corresponding z-score is approximately 1.645, which adjusts the width of the confidence interval to achieve the stated level of certainty.

Maximum Variance and Worst-case Estimate

When the true population proportion is unknown, the maximum variance occurs when the proportion is 0.5, because p(1-p) is maximized at this value. By using 0.5 as the estimate, the calculation ensures that the sample size will be sufficiently large regardless of the actual proportion, providing a conservative, worst-case estimate.

Margin of Error

The margin of error defines the maximum expected difference between the true population parameter and the sample estimate. A margin of error of 0.03 indicates that the estimate should be within plus or minus 3 percentage points of the true proportion, directly influencing the precision of the sample size determination.

Confidence Level

A confidence level represents the degree of certainty that the true population parameter (in this case, the percentage of Republicans among registered voters) falls within the estimated interval. A 90% confidence level means that if the sampling process were repeated many times, about 90% of the constructed intervals would contain the true value, reflecting how certain we want to be about our estimation.

Sample Size Determination for Proportions

This concept involves calculating the number of observations required to estimate a population proportion within a specified margin of error and confidence level. The formula used incorporates the margin of error, the critical value from the normal distribution, and an estimate of the variability of the proportion.

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