ECON 221 Assignment 3

1. The scores of all
applicants taking an aptitude test required by a law school follow a normal
distribution with mean 420 and standard deviation 100. (20 points)

a.
What is
the probability that the mean score of a sample of 25 applicants will be between
400 and 450? (4 pts)

b.
There is a 10% probability that the
mean score of a random sample of 25 applicants is higher than what number? (4 points)

c.
There is a
75% probability that the mean scoreof a random sample of 25
applicantsis less than what
number? (4 points)

d.
If the
sample standard deviation is larger than the population standard deviation, there
is a 2.5% probability that the standard deviation of the scores of a sample of
25 applicants is larger than what number?
(4 points)

e.
If the
sample standard deviation is smaller than the population standard deviation,
there is a 2.5% probability that the standard deviation of the scores of a
sample of 25 applicants is less than what number? (4 points)

2. In a particular
year, 40% of home sales were partially financed by the seller. A random sample of 250 sales was
examined. (16 points)

a.
What is the probability that 35% of
them, or less, would be partially financed by the seller? (4 points)

b.
There is 80% chance that the sample
proportion of home sales that would be partially financed by the seller will be
bigger than what amount? (4 points)

c.
There is 90% chance that the sample
proportion of sales that would be partially financed by the seller will be less
than what amount? (4 points)

d.
There is 70% probability that the
sample proportion will differ from the population proportion by up to a maximum
of how much? (4 points)

3.
The scores of applicants
taking an aptitude test necessary for entrance exams follow a normal
distribution. A random sample of 25 scores
is taken. (16 points)

a.
What is the probability
that the sample variance is up to 50% larger than the population variance? (4 points)

b.
What is the probability
that the population variance is up to 50% larger than the sample variance? (4 points)

c.
There is a 95% probability
that the population variance is smaller than the sample variance to an amount
of ____% of the sample variance. (4 pts)

d.
There is a 90% probability
that the population variance is larger than the sample variance up to an amount
of____% of the sample variance. (4 points)

4. At the beginning of the course of
Statistical Methods I, the instructor recommended that students devote 3 hours
per week for the duration of the 13-week semester, for a total of 39 hours. It
is known that the times spent on studying statistics follow a normal
distribution. Throughout the course’s
duration, students were claiming that they were following the instructor’s
recommendation. At the course’s
completion, a random sample of 9 students enrolled in the course was drawn
where each student was asked how many hours he or she spent doing homework in
statistics. The data are listed
below: (20 points)

45 38 37 40 44 38 46 38 43

a. Calculate the sample mean and sample
standard deviation. (5 points)

b. Estimate
the population standard deviation with 90% probability. (5 points)

c. If
the same sample results had been obtained from a random sample of 16 students,
would your conclusion in (b) about the students’ time spent be different?Calculate and explain. (3 points)

d. Using the original data, build a 95% confidence
interval for the mean time spent by the students in studying. Does it include the instructor’s suggestion?
(3 points)

e. If
the sample had been drawn from a class that had 60 students altogether, would
your findings in (d) about the range of themean time spent by the students in
studying be different? Calculate and explain. (4 points)

5. A pharmaceutical
company wants to test the effectiveness of an experimental sleeping pill. An experiment is conducted over two nights,
with 9 volunteers who have difficulties sleeping at night. On the first night, each of the ten persons
is given the sleeping pill one hour before bedtime and the time needed to fall
asleep is recorded. On the second night,
the same persons are given a placebo in the form of a pill that looks exactly
as the real one (without being told about it) and the time needed to fall
asleep is again recorded. The times
needed to fall asleep are assumed to follow a normal distribution. The results for the two nights for the nine
volunteers are shown below: (10 points)

Individual

A

B

C

D

E

F

G

H

I

Pill

4

7

10

15

6

12

10

1

11

Placebo

7

9

8

16

10

15

15

4

10

a. Calculate the mean and
variance of the sample data. (6 points)

b. Estimate with 95%
confidence the mean reduction in time it takes for someone who has taken the
sleeping pill to fall asleep. (4 points)

6.
Assume
that the distribution of household income of the people playing at the casino
is normally distributed with a standard deviation of $6,800. Suppose that for a sample of 25 households,
it is found that the average income was $38,200. (8 points)

a.
Based
on the sample findings, estimate the true mean income of casino players with
90% probability. (4 points)

b.
If we
wanted to shorten the interval of the estimation of the true mean income of
casino players to awidth of $1000 in total, what should the size of our
sample be to achieve it? (4 points)

7.
A
clinic offers a weight-loss program. A
review of its records found the following amounts of weight loss in pounds for
a random sample of 16 of its clients at the conclusion of a 4-month program: (10 points)
18 25 16
11 15 20 16 19
28 25 26
31 42 38 35
19

Calculate the mean and variance of the
sample. (4 points)

Build a 99% confidence interval for
the mean weight loss of the people going through this program. (3 points)

c. If the total number of
people who had registered for this weight-loss program had been 100, would your
conclusions in (b) above be different?
Calculate and explain. (3 points)