A random sample of size 64 is to be used to test thenull hypothesis that for a certain age group the mean score on an achievement test (the mean of a normal pop-ulation with σ2 = 256) is less than or equal to 40.0 against the alternative that it is greater than 40.0. If the nullhypothesis is to be rejected if and only if the mean of therandom sample exceeds 43.5, find (a) the probabilities of type I errors when μ = 37.0, 38.0,39.0, and 40.0;(b) the probabilities of type II errors when μ = 41.0, 42.0,43.0, 44.0, 45.0, 46.0, 47.0, and 48.0.Also plot the power function of this test criterion.

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2-Proportion Test

In a 2-proportion test, two separate proportions are measured at the same time, and their similarity is evaluated. It is also known as a 2-proportion z-test, and a comparison of two different proportions is done in this method.