Confidence Intervals

Summary :

This chapter uses business examples, such as estimating monthly iTunes downloads from a marketing survey, to introduce confidence intervals. It explains how a point estimate becomes an interval estimate, covers the Student's t-distribution for unknown standard deviations, and shows how to size a sample for a target confidence level and margin of error.

From point estimate to confidence interval

A sample mean or sample proportion gives a single point estimate of an unknown population parameter, but a confidence interval expresses that estimate as a range together with a stated probability of accuracy, called the confidence level. Using a marketing example of estimating the mean number of songs downloaded per month, the chapter shows how the central limit theorem lets an analyst attach two standard deviations to a sample mean to state, for instance, 95 percent confidence that the true population mean falls within a given interval.

Known standard deviation and large samples

When the population standard deviation is known, or the sample is large, the confidence interval for a population mean is built directly from the normal distribution and the standard error of the sampling distribution of means. The width of the interval depends on the desired confidence level, set by the z-value the analyst chooses, and on the sample size, since a larger sample narrows the interval for the same confidence level.

Business applications of the interval framework

Because managers rarely know a population's exact standard deviation, the chapter extends the framework to the Student's t-distribution, which adjusts for the extra uncertainty of estimating the standard deviation from the sample itself, and shows how to work backward from a desired margin of error and confidence level to the sample size a survey or study needs before it is run.