CATCHING UP. THINGS WE SHOULD KNOW:
PROBLEM SET #1
These are some of the concepts you should have grasped from QUME 232. Answer as
many as you can. We’ll be reviewing these in the next few classes.
Q.1
The summation sign. Interpret (simplify) the following:
4
x
a)
i 0
i 2
2
 (i  5)
b.)
i 0
5
3
c.)
i 3
i 3
 x y 
4
d.)
3
i 2 j 2
2
i
j
2i
 4i  3
e.)
i 1
Q.2
In September, Nanaimo’s daily high temperature has a mean of 70oF and a standard
deviation of 7oF. What are the mean, standard deviation and variance in oC (celcuis)?
5
Hint: C =(9) (𝐹 − 32)
Q.3
The following table gives the joint probability distribution between employment status
and college graduation among those either employed or looking for work (unemployed)
in the working age US population 2008
Non-college grads (X=0)
College grads (X=1)
Total
a.
b.
Unemployed (Y=0)
0.037
0.009
Employed (Y=1)
0.622
0.332
Total
Compute E(X) and E(Y)
The unemployment rate is a fraction of the labor force that is unemployed.
Show that the unemployment rate is given by 1 - E(Y)
c.
d.
e.
Calculate E(Y|X=1)
A randomly selected member of this population reports being unemployed.
What is the probability that this worker is a college graduate?
What is the covariance between educational achievement and employment
status? What does this suggest about the independence or dependence of
the two variables?
Q.4
In a given population of two-earner male/female couples, male earnings have a mean
of $40,000 per year and a standard deviation of $12,000. Female earnings have a mean
of $45,000 per year and a standard deviation of $18,000. The correlation between male
and female earnings for a couple is 0.80. Let C denote the combined earnings for a
randomly selected couple
a.
What is the mean of C
b.
What is the covariance between male and female earnings?
c.
What is the standard deviation of C
Q.5
Suppose you have some money to invest—for simplicity, $1—and you are planning to
put a fraction α into a stock market mutual fund and the rest, 1-α, into a bond mutual
fund. Suppose that a $1 invested in a stock fund yields Rs return after 1 year and that a
$1 invested in a bond fund yields Rb. Suppose that Rs is random with a mean of 0.08
(8%) and a standard deviation of 0.07, and suppose Rb is random with a mean of 0.05
(5%) and a standard deviation of 0.04. The correlation between Rs and Rb is 0.25. If you
place a fraction α in the stock fund and 1-α in the bond fund, the return on your
investment R = αRs + (1-α)Rb
a. Suppose α = 0.5 compute the mean and standard deviation of R
b. (Harder) Standard deviation can be used to measure the risk of an investment.
What is the value of α that minimizes the standard deviation (risk) of the return
on your investment R (Show using a graph, algebra or calculus)
c.
Q.6 (Multiple Choice)
i.
An estimator ˆY of the population value Y is unbiased if
a. ˆY  Y .
b. Y has the smallest variance of all estimators.
p
c. Y  Y .
d. E(ˆY )  Y .
ii.
The critical value of a two-sided t-test computed from a large sample
a. is 1.64 if the significance level of the test is 5%.
b. cannot be calculated unless you know the degrees of freedom.
c. is 1.96 if the significance level of the test is 5%.
d. is the same as the p-value
iii.
A type II error
a. is typically smaller than the type I error.
b. is the error you make when choosing type II or type I.
c. is the error you make when not rejecting the null hypothesis when it is
false.
d. cannot be calculated when the alternative hypothesis contains an “=”.
v.
A large p-value implies
a. rejection of the null hypothesis.
b. a large t-statistic.
c. a large Y act .
d. that the observed value Y act is consistent with the null hypothesis
Q.7
A woman reported being in labor for 72 hours before giving birth. Assuming that labor
before delivery follows a normal distribution with a mean of 60 hours and a variance of
25 hours; would you conclude that it is unusual for a woman to be in labor for more
than 72 hours? Explain using probability values.
Q.8
In a population with mean 100 and variance 43, what is the probability that in a random
sample of 165 items from this population, the sample mean will exceed 98?
Q.9
A new version of the Math Provincial Exam is given to 1000 randomly selected high
school seniors. The sample mean score is 1110 and the sample standard deviation is
123. Construct a 95% confidence interval for the population mean test scores for high
school seniors.
Q.10
To investigate possible gender discrimination in a firm, a sample of 100 men and 64
women with similar job descriptions are selected at random. A summary of the
resulting monthly salaries follows:
.
mean
3100
(a)
Men
Standard
deviation
200
n
mean
100
2900
Women
Standard
deviation
320
n
64
Do these data represent statistically significant evidence that average wage of
men and women are different? Test the hypothesis at the 5% significance level
Q.11
Math SAT scores (Y) are normally distributed with a mean of 500 and a standard
deviation of 100. An evening school advertises that it can improve students’ scores by
roughly a third of a standard deviation, or 30 points, if they attend a course which runs
over several weeks. (A similar claim is made for attending a verbal SAT course.) The
statistician for a consumer protection agency suspects that the courses are not
effective. She views the situation as follows: H 0 : Y  500 vs. H1 : Y  530 .
(a)
(b)
(c)
Sketch the two distributions under the null hypothesis and the alternative
hypothesis.
The consumer protection agency wants to evaluate this claim by sending 50
students to attend classes. One of the students becomes sick during the course
and drops out. What is the distribution of the average score of the remaining 49
students under the null, and under the alternative hypothesis?
Assume that after graduating from the course, the 49 participants take the SAT
test and score an average of 520. What is the p-value for such a score under the
null hypothesis? What would you conclude if your significance level is 5% based
on this p-value?
Q.12
How many of the questions, including this one, were you able to answer with little or no
difficulty?