During routine screening, a doctor notices that 22 % of her...

During routine screening, a doctor notices that $22 \%$ of her adult patients show higher than normal levels of glucose in their blood - a possible warning signal for diabetes. Hearing this, some medical researchers decide to conduct a large-scale study, hoping to estimate the proportion to within $4 \%$ with $98 \%$ confidence. How many randomly selected adults must they test?

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Key Concepts

Z-score / Critical Value

The z-score, or critical value, corresponds to the desired confidence level and reflects the number of standard deviations a point is from the mean in a standard Normal distribution. It is used to scale the standard error in sample size calculations and in constructing confidence intervals.

Sample Size Calculation

Sample size calculation involves determining the number of observations required to estimate a parameter with a specified margin of error and confidence level. For estimating proportions, this typically requires the use of a formula that incorporates the estimated proportion, the desired margin of error, and the appropriate z-score corresponding to the confidence level.

Confidence Level

The confidence level represents the degree of certainty you want in your estimate of the population parameter. In the context of confidence intervals, it indicates the percentage of all possible samples that can be expected to include the true parameter value if the same study were repeated multiple times.

Margin of Error

The margin of error is the maximum expected difference between the true population parameter and a sample estimate. It defines the precision of the estimate and is crucial in determining the necessary sample size to achieve a desired level of accuracy.

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