Suppose a statistics professor offers two midterm exams, each worth 100 points. A student plans to do a lot of traveling during the semester for job interviews, and asks if she can miss one of the midterms. The professor gives her the option of taking both midterms as scheduled, or taking just one midterm and doubling the score.
a. Which of the random variables 2X1 and X1 + X2 represents the student's score if she takes both midterms? Which represents her score if she takes just one midterm and doubles the score?
b. If the student takes both midterms, and does somewhat worse than usual on one, she may do somewhat better than usual on the other, so that her combined midterm score isn't too bad. If she takes just one midterm, and does somewhat worse than usual, then the doubled score is also worse than usual; if she does somewhat better, then the doubled score is also better than usual. In which case is there more of a tendency for unusually low or high scores-if she takes both midterms, or if she takes just one midterm and doubles the score?
c. This is a fairly average student, and her exam performance Xi is a random variable with mean 80. Find the mean of 2X1 and find the mean of X1 + X2.
d. Are the means you found in part (c) equal? If not, tell which is larger and why.
e. Suppose a student's exam performance Xi has standard deviation 5. Find the standard deviation of 2X1 and find the standard deviation of X1 + X2, according to formulas on page 282 and 284.
f. Are the standard deviations you found in part (e) equal? If not, tell which is larger and why.
g. To find the standard deviation of X1 + X2 in part (e), one must assume that a student's scores X1 and X2 on the first and second midterm exams are independent. Is this a reasonable assumption?