Question to solve with either Excel or R. Suppose Mark is taken to the emergency room after a car wreck and the doctors think he might have a brain injury. They give him a finger-tapping-speed test that has a mean of 80 and a standard deviation of 15 in the normal population and they decide to use a one-tailed test with all 5% of alpha in the lower end of the distribution. That means that they will conclude that he has brain damage if his score is at or below the point in the distribution that marks the lower 5% of the distribution. Also, suppose that the doctors believe that they know that the mean of the alternative distribution is 65 with a standard deviation of 15.  Use Excel or R to find the probability of making a Type II error in this scenario.

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The video and the textbook describe two steps to find type II error and power. First, you find the X value that marks the point at which you will reject the null hypothesis in the null hypothesis testing situation. The null hypothesis used a distribution with a mean of 80 and SD of 15 and included all 5% of alpha in the left tail of the distribution (same as the video and the textbook). The X value is 55.3272 and you find that in Excel with NORM.INV((0.05,80,15). In R, you find it with qnorm(0.05,80,15).

 

The second step involves the alternative distribution, which had a mean of 65 and SD of 15. The video and textbook explain that anything below the X value (i.e. 55.3272 in this case) in the alternative distribution is power because that is the area in which you correctly reject the null hypothesis. The area to the right of that point is Type II error because that is the area in which you incorrectly fail to reject the null hypothesis. In a recent version of Excel, you get this by

1-NORM.DIST(55.3272,65,15,TRUE) and the R code is 1-pnorm(55.325,65,15).