A critical region for a known variance

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Systolic blood pressure in a reference population has mean 127 and standard deviation 7. A sample of 101 people from another group has sample mean \overline X. Assume the readings are independent and normal with that same standard deviation, and mean \mu. The test statistic is

T=\frac{\overline X-127}{7/\sqrt{101}}.

Find the two-sided critical region for T, and the corresponding region for \overline X, at significance level 0.05. Use z_{0.975}=1.960.

Worked answer (click to show)

Under H_0:\mu=127, T is standard normal. The two-sided level-0.05 region is |T|\ge 1.960. Because the standard error is 7/\sqrt{101}\approx 0.6965,

|\overline X-127|\ge 1.960\times 0.6965\approx 1.365.

Reject when \overline X\le 125.635 or \overline X\ge 128.365. The observed mean of 130 is well inside the region.