Question 1 Problem Set The National Assessment of Educational Progress (NAEP) is

Question 1
Problem Set
The National Assessment of Educational Progress (NAEP) is a national program that provides assessments
of students at different grade levels and different subjects. Suppose that the scores for twelfth grade students
in mathematics is approximately N(288,38).
(a) How high a score is needed to be in the top 25% of twelfth graders who take this exam?
(b) How high a score is needed to be in the top 5%?
Question 2
Typographical and spelling errors can be either “nonword errors” or “word errors.” A nonword error is not a
real word, as when “the” is typed as “teh.” A word error is a real word, but not the right word, as when
“lose” is typed as “loose.”
When undergraduates are asked to write a 250-word essay (without spell-checking), the number of nonword
errors has the following distribution:
Errors Probability
0 0.1
1 0.3
2 0.3
3 0.2
4 0.1
Table 1: Distribution of nonword errors
The number of word errors has this distribution:
Errors Probability
0 0.4
1 0.3
2 0.2
3 0.1
Table 2: Distribution of word errors
(a) Let X denote the nonword errors and Y denote the word errors. What is E(X + Y )?
(b) Assuming X and Y are independent, what is Var(X + Y )?
(c) If ρXY = 0.4, what is Var(X +Y)?
Question 3
Suppose weights of the checked baggage of airline passengers follow a nearly normal distribution with mean
45 pounds and standard deviation 3.2 pounds. Most airlines charge a fee for baggage that weigh in excess of
1
50 pounds. Determine what percent of airline passengers incur this fee.
Question 4
Find the standard deviation of the distribution in the following situations:
(a) MENSA is an organization whose members have IQs in the top 2% of the population. IQs are normally
distributed with mean 100 and the minimum IQ score required for admission to MENSA is 132.
(b) Cholesterol levels for women aged 20 to 34 follow an approximately normal distribution with mean
185 milligrams per deciliter (mg/dl). Women with cholesterol levels above 220 mg/dl are considered to
have high cholesterol and about 18.5% of women fall into this category.
Question 5
To estimate the mean height μ of male students on your campus, you will take a SRS of students. Heights
of people of the same sex and similar ages are close to Normal. You know from government data that the
standard deviation of the heights of young men is about 2.8 inches. Suppose that (unknown to you) the mean
height of all male students is 70 inches.
(a) If you choose one student at random, what is the probability that he is between 69 and 71 inches tall?
(b) You measure 25 students. What is the sampling distribution of their average height x ̄?
(c) What is the probability that the mean height of your sample is between 69 and 71 inches?