If you wish to estimate a population mean to within .2 with...

If you wish to estimate a population mean to within .2 with a $95 \%$ confidence interval and you know from previous sampling that $\sigma^{2}$ is approximately equal to 5.4 , how many observations would you have to include in your sample?

Key Concepts

Population Variance

Population variance is a measure of the spread of data points in a population around the mean. When the variance is known or can be reasonably estimated from previous data, it allows for the use of the standard normal distribution (z-distribution) in constructing confidence intervals and hence simplifies the sample size determination process for estimating the mean.

Sample Size Calculation

Sample size calculation for estimating a population mean with a specified margin of error involves rearranging the formula for the margin of error. By setting the margin of error equal to the product of the critical value and the standard error (which involves the population standard deviation divided by the square root of the sample size), one can solve for the minimum required sample size needed to achieve the desired precision.

Margin of Error

The margin of error defines the maximum expected difference between the true population parameter and a sample estimate. In the context of estimating a mean, it is calculated as the product of a critical value and the standard error, and it sets the precision of the estimate by specifying how close the sample estimate should be to the actual parameter.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter, such as the mean. It provides an estimate of the uncertainty around a sample statistic by indicating a level of confidence (for example, 95%) that the interval captures the true parameter.

Standard Normal Distribution

The standard normal distribution, a normal distribution with a mean of 0 and a standard deviation of 1, is used to obtain critical values (z-scores) in confidence interval calculations when the population variance is known. The appropriate z-score corresponding to the desired confidence level (such as 1.96 for 95% confidence) is used to determine how wide the interval should be.

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