Homework 5 Solution

chapter 13, #34

a) There are several reasons:
1) There is no comparison group. Perhaps *everyone* who takes the exam twice goes up by 60 points or so.
2) This is an observational study. We don't know how the 30 students came to enroll in this program, but it does not sound like they were randomly selected or were randomized to participate in this program or a "control" program. And so there could be confounding variables -- perhaps the factors that make a student enroll in such a program also make them the sort of student who improves dramatically the second time they take an exam, regardless of their preparation for the second attempt.

Note: for full credit you need detailed responses.

b) We need a "control" program in which students are given no clear advice on how to prepare for the SAT. Then we take a set of students who have already taken the exam once. Half are randomly assigned to this study course, the others to our control course. This would be a controlled experiment with randomized design and a placebo. We should also make sure that the researchers not be the ones teaching the course, so that they can't influence the students' performance in any way.

c) In this case we might stratify and make sure that low-scoring students were approximately equally represented in both the control and the treatment group.

Chapter 14, #25

a) Depends on the day. Some days have more than 3 digits. (October 31, for example). There are 65 such days. So for those days its impossible. For the other days, there are 1000 possible outcomes (000 through 999), and so the chance is 1/1000.
b) Well, there are 65 days we dont' have to worry about,a nd 300 we do. For each of the 300 days, the probability of it NOT happening is 999/1000 or .999. So P(it doesn't happen all year) = P(doesn't happen Day 1 AND doesn't happen Day 2 AND .... Day 300). Since these trials are independent (that's how the lottery is designed!), we can use the multiplication rule:
= P(not on Day 1) * P(not on day 2) *.....*P(not on day 300) = .999*.999*..... *.999 = (.999) raised to the 300th power or : 0.741

c) P(match at least once) = 1 - P(never match) = 1 - .741 = .259

d) P(at least one of the 50 gets 911) = 1 - P(none gets 911) = 1 - (.999)⁵⁰ = .0487