It’s believed that as many as 25% of adults over 50 never...

It’s believed that as many as 25% 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 6% 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 3%?

It’s believed that as many as 25% 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 6% 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 3%?

Key Concepts

Population Proportion Estimation

When planning a survey to estimate a proportion, one typically uses a preliminary estimate or assumed value for the population proportion. This assumed value is used in the sample size formula to calculate the variability and hence determine the required sample size to achieve the desired precision.

Sample Size Determination for Proportions

Sample size calculation for proportions involves determining the number of observations required so that the estimated proportion falls within a desired margin of error at a specified confidence level. This calculation is crucial for ensuring the reliability and accuracy of survey results while avoiding unnecessary data collection.

Z-Score

The Z-score is a critical value from the standard normal distribution corresponding to a given confidence level. It is used in the formula for determining the margin of error and sample size, serving as a multiplier that adjusts for the level of certainty desired in the estimation.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate. It is influenced by the confidence level and the variability in the data, and defines the precision of the estimate within the confidence interval.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter at a specified confidence level. For proportions, it allows us to estimate the true proportion of a characteristic in a population with a given level of certainty.

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