Find the indicated sample size. A researcher wants to...

Find the indicated sample size. A researcher wants to estimate the mean grade point average of all current college students in the United States. She has developed a procedure to standardize scores from colleges using something other than a scale between 0 and $4 .$ How many grade point averages must be obtained so that the sample mean is within 0.1 of the population mean? Assume that a $90 \%$ confidence level is desired. Also assume that a pilot study showed that the population standard deviation is estimated to be 0.88.

Find the indicated sample size.
A researcher wants to estimate the mean grade point average of all current college students in the United States. She has developed a procedure to standardize scores from colleges using something other than a scale between 0 and $4 .$ How many grade point averages must be obtained so that the sample mean is within 0.1 of the population mean? Assume that a $90 \%$ confidence level is desired. Also assume that a pilot study showed that the population standard deviation is estimated to be 0.88.

Key Concepts

Confidence Interval

A confidence interval for a population mean is a range of values, derived from sample statistics, that is likely to contain the true population mean with a specified level of confidence. It is determined by the sample mean, the standard deviation, and a critical value that corresponds to the desired confidence level.

Margin of Error

The margin of error is the amount added and subtracted from the sample mean to create the confidence interval. It represents the maximum expected difference between the sample mean and the true population mean, and is influenced by the standard deviation, the sample size, and the critical value from the chosen confidence level.

Critical Value

The critical value is a point on the standard normal distribution (or other distribution) that corresponds to the desired level of confidence in a confidence interval. For a given confidence level, it defines the number of standard deviations from the mean required to capture the central percentage of the distribution.

Standard Deviation

Standard deviation is a measure of dispersion or variability within a set of data. In the context of estimating a population mean, the population standard deviation is used to quantify the spread of the data and impacts the width of the confidence interval and the required sample size.

Sample Size Calculation

Sample size calculation involves determining the number of observations needed for a study to achieve a desired margin of error at a specific confidence level. The formula, n = (critical value * standard deviation / margin of error)^2, is used to ensure that the sample mean will be within a specified distance of the true population mean.

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