Page 1
Statistics 21
Problems from past Fnal exams.
1.
(10 points) A psychologist administrated a multiple-choice test to 122 college students. Each
question had Fve possible answers and only one was correct. The questions were taken from the
reading comprehension section of the SAT verbal test, with one important change. The SAT pres-
ents short passages to read and follows these with multiple-choice questions about the meaning
of the passages. ±or his test, the psychologist took only the questions–not the reading passages.
So someone taking the psychologist’s test has to answer the SAT questions without having read
the passages. Why make up such a crazy test? The psychologist was trying to Fnd out whether
this part of the SAT was really testing reading ability or some other test-taking skill. None of
the 122 students had taken the SAT. Their average score on the psychologist’s test turned out to
be 38 points out of 100. (There were 100 questions; a correct answer was worth one point and an
incorrect answer zero points.) The psychologist took the average of 38 as evidence that points can
be earned on this part of the SAT by using mental skills other than the ability to understand the
passages. Of course, the psychologist realized that the students would get some answers right
just by luck.
(a) ±ormulate the null hypothesis implicit in the above paragraph as a statement about a box
model.
(b) Calculate the appropriate test statistic.
(c) Would you reject the null hypothesis?
2.
(10 points) A restaurant menu offers four main courses priced $7, $10, $12, or $15, and three
desserts priced $2, $3, or $4. A regular customer always tips at one of the three rates 10%, 15%,
or 20%. On one occasion, the customer decides to choose his main course, his dessert, and his
tipping rate at random. Let X be the price of his main course, Y the price of his dessert, and T the
tipping rate.
You may assume the three successive choices (of main course, dessert, tipping rate) are made
independently of one another. Also, to choose “at random” is to give all the possibilities an
equal chance of being chosen.
(a) Let U be the price of the food, excluding tip. Write down an equation for U in terms of
X, Y.
(b) ±ind the expected value and the SE of U.
(c) Let V be the price of the meal, including tip. Write down an equation for V in terms of X, Y
and T.
(d)
±ind the expected value of V.
3.
(5 points) A person will be picked at random from the class in problem #23 and you will be told
his or her midterm score. Using this information, you have to guess his or her Fnal score. Some-
one says:
“Look, the Fnal average was exactly twice the midterm average. So, just
double the midterm score you are told and use that as your guess.