Confidence Interval for a Mean



(8.3) Confidence Interval for a Mean Practice

The purpose of a confidence interval is to estimate an unknown population parameter using a sample statistic, with an indication of how accurate the estimate is (+/–) and of how confident we are that the result is correct (95%).

z-interval for a population mean: If a SRS of size n is drawn from a normally distributed population having an unknown mean μ and known standard deviation σ, a level C confidence interval for μ is

If the sample size n is large, this formula is approximately correct even when the population does not have a normal distribution.

ex. IQ tests are normed (normally distributed) with a μ = 100 and σ = 15. A random sample of 100 students from a high school has a mean IQ score of 112 and standard deviation of 16. Calculate a 95% CI for IQ scores at this school.

Check using 7:ZInterval on your calculator.

t-interval for a population mean: In reality, you usually do not have the population standard deviation (if you know the SD, you usually also know the mean…) You can use a

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The t-distribution changes as the sample size changes—the t-table uses

_________________________________. If the needed df is not in the table, use the next smaller degrees of freedom to give a larger, more conservative CI. t is a family of curves—as n gets larger, the curve gets closer to normal.

ex. Questionnaires were sent to a SRS of 160 major U.S. hotel chain managers and 114 responses were returned. The average reported time that the managers had spent with their current company was 11.78 years with standard deviation of 3.2 years. Give a 99% confidence interval for the time spent by all U.S. hotel chain managers.

Check using 8:TInterval on your calculator—calculator uses a huge formula to calculate degrees of freedom (gives a larger df allowing for a smaller interval). Don’t forget to include this df if using the calculator interval.

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