13.50 Tests from confidence intervals. A confidence in terval for the population mean fx tells us which values of jx 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 Hq: fx = fxo starting with a confidence interval: re ject Ho if fXo is outside the interval and fail to reject if fXo is inside the interval.

The alternative hypothesis is always two-sided, Ha', fx ^ fXQ, because the confidence interval extends in both directions from X. A 95% confidence interval leads to a test at the 5% signif icance level because the interval is wrong 5% of the time. In general, confidence level C leads to a test at significance level a = 1 - C.

(a) In Example 13.10, 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 ct= 15.

Give a 90% confidence interval for the mean blood pressure fx of all executives.

(b) The hypothesized value fio = 128 falls inside this confi dence interval. Carry out the z test for Hq: ^ = 128 against the two-sided alternative. Show that the test is not significant at the 10% level.

(c) The hypothesized value fXo = 129 falls outside this confi dence interval. Carry out the z test for Hq: fx = 129 against the two-sided alternative. Show that the test is significant at the 10% level.