Question

Out of 32 confidence intervals, we would expect that 24 confidence intervals would contain the population mean for a 75% confidence interval, that 27.2 confidence intervals would contain the population mean for an 85% confidence interval, and that 30.4 confidence intervals would contain the population mean for a 95% confidence interval. However, in reality, only 26 confidence intervals contain the population mean for 75%, 27 intervals for 85%, and 30 intervals for 95%.

a.) How do you account for the discrepancies between the expected number of confidence intervals and the actual number of confidence intervals containing the population mean?

b.) What can be done so that the expected number of confidence intervals containing the population mean and the actual number of confidence intervals containing the population mean are likely to be closer? Explain your reasoning.

Answer 1

a)

The discrepancies between the expected number of confidence intervals and the actual number of confidence intervals containing the population mean is the low number of confidence intervals (32) used. With low number of confidence intervals, the standard error of actual number of confidence intervals containing the population mean will be higher and thus there will be large variation in actual number of confidence intervals which led to considerable difference between expected and the actual number of confidence intervals containing the population mean.

b)

The standard error of  actual number of confidence intervals containing the population mean is,

where p is the confidence interval and n is the number of confidence intervals used. We see that the standard error of actual number of confidence intervals containing the population mean will decrease with increase in number of confidence intervals used. For a higher n, the expected number of confidence intervals containing the population mean and the actual number of confidence intervals containing the population mean are likely to be closer.