It's believed that as many as 21 % of adults over 50 never...

It's believed that as many as $21 \%$ of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group. a) How many of this younger age group must we survey in order to estimate the proportion of non-grads to within $10 \%$ with $90 \%$ confidence? b) Suppose we want to cut the margin of error to $4 \%$. What's the necessary sample size? c) What sample size would produce a margin of error of 5\%?

It's believed that as many as $21 \%$ of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group.
a) How many of this younger age group must we survey in order to estimate the proportion of non-grads to within $10 \%$ with $90 \%$ confidence?
b) Suppose we want to cut the margin of error to $4 \%$. What's the necessary sample size?
c) What sample size would produce a margin of error of 5\%?

Key Concepts

Confidence Intervals

A confidence interval is a range of values used to estimate an unknown population parameter, constructed so that it contains the parameter with a specified probability (the confidence level). In the context of proportions, the interval is calculated around an estimated percentage, providing an understanding of the precision and reliability of the estimate made from sample data.

Margin of Error

The margin of error quantifies the amount of random sampling error in the estimate of a population parameter. It defines the maximum expected difference between the true population proportion and the sample proportion, and is calculated using a critical value and the standard error of the estimate. A smaller margin of error indicates a more precise estimate.

Sample Size Calculation for Proportions

This concept involves determining the number of observations required so that the estimated proportion falls within a desired margin of error with a specified level of confidence. The calculation uses the critical value from the corresponding confidence level, the estimated proportion, and the target margin of error. The formula ensures that the sample provides enough information to reliably estimate the population proportion.

Critical Values (Z-scores)

Critical values, particularly Z-scores in the context of normal approximations, are used to determine the number of standard deviations a point is from the mean. In confidence interval calculations for proportions, the Z-score corresponding to the desired confidence level (for instance, 1.645 for 90% confidence) is used to scale the standard error, playing an essential role in computing the margin of error and the necessary sample size.

Transcript

Please give Ace some feedback