1. Suppose reaction time in some population is Normally distributed with mean 470 and standard deviation 40 (both measured in milliseconds). We randomly select a person from this population, and measure the reaction time.
a. Find the probability the reaction time is 500 or less.
b. Find the probability the reaction time is at least 400.
c. Find the probability the reaction time is between 450 and 550.
d. Find the 96th percentile of the distribution of reaction time (at least approximately). That is, find the value such that about 96% of the distribution is below this point.
2. Suppose women with anorexia have a daily caloric intake that is Normally distributed with mean 1385 and standard deviation 65. Further, suppose women without anorexia have a daily caloric intake that is Normally distributed with mean 1720 and standard deviation 110. We use a single day's caloric intake as a screening test for anorexia. We decide a value under 1500 is a positive screen, so a value over 1500 is a negative screen.
a. Briefly explain why a relatively low value of this variable is a positive screen.
b. Calculate the sensitivity of the test. This is the probability of having a positive screen, given that the person has the disease, so use the distribution for women with anorexia.
c. Calculate the specificity of the test. This is the probability of having a negative screen, given that the person does not have the disease, so use the distribution for women without anorexia.
d. Suppose we changed the cutoff point from 1500 to 1600. Would the sensitivity increase or decrease? You don't have to make any calculations if you can answer this without doing so.
e. Suppose we changed the cutoff point from 1500 to 1600. Would the specificity increase or decrease? You don't have to make any calculations if you can answer this without doing so.
f. Briefly describe how we could decide on the "best" cutoff point.