1-A continuous random variable X has a uniform distribution over the interval (-k, 3k), where k is some number. The height of the PDF over this interval is .05. For all other values of X, the PDF=0.
a.) What is the value of k?
b.) What is the probability that X=7.5?
c.) What is the probability that a randomly selected X lies between -10 and 10?
2-Suppose a selective university only considers accepting students with a cumulative grade point average of 3.16 or higher. Students below this threshold are not considered. Suppose the population of students applying to this university has a GPA that is normally distributed with a mean of 3.0 and standard deviation .4.
a) What is the probability that a randomly selected student will meet the GPA threshold?
b) If 50 applicants are chosen at random, what is the probability that 17 or more of them will meet the GPA threshold? (Note: You’ll want to use your answer from part a. If you’re not sure about this answer (and even if you are), be very clear about your calculations for part b.)
3-Suppose 76 stock market mutual funds are chosen at random and their results for the most recent year are recorded. Of these, after fees and expenses, 18 of them earned their investors a return higher than the stock market as a whole. (Note for those interested in investing: this is a pretty realistic figure. Most mutual funds fail to overperform the broader market in a year.)
What is the 97% confidence interval for the population proportion of mutual funds that have a return higher than the stock market as a whole?