Tests from confidence intervals. A confidence interval for the population mean tells us which values of are plausible (those inside the interval) and

Tests from confidence intervals. A confidence interval for the population mean

μ tells us which values of μ are plausible (those inside the interval) and which values are not plausible (those outside the interval) at the chosen level of confidence.

You can use this idea to carry out a test of any null hypothesis H0: μ = μ0 starting with a confidence interval: reject H0 if μ0 is outside the interval and fail to reject if μ0 is inside the interval.

The alternative hypothesis is always two-sided, Ha: μ = μ0, because the confidence interval extends in both directions from x. A 95% confidence interval leads to a test at the 5% significance level because the interval is wrong 5% of the time.

In general, confidence level C leads to a test at significance level α = 1 − C.

(a) In Example 14.9, a medical director found mean blood pressure x = 126.07 for an SRS of 72 executives. The standard deviation of the blood pressures of all executives is σ = 15. Give a 90% confidence interval for the mean blood pressure μ of all executives.

(b) The hypothesized value μ0 = 128 falls inside this confidence interval. Carry out the z test for H0: μ = 128 against the two-sided alternative. Show that the test is not significant at the 10% level.

(c) The hypothesized value μ0 = 129 falls outside this confidence interval. Carry out the z test for H0: μ = 129 against the two-sided alternative. Show that the test is significant at the 10% level.