STAT 201:23
Assignment 3 on 5.1, 5.2, 5.3 and 5.4
The following questions are to be answered and handed in (on paper) by the end of class on
Wednesday, March 9.
1. (5.3 page 272) Consider the population distribution shown below.
x
1
2
3
4
5
p(x)
0.2
0.3
0.2
0.2
0.1
The random variable x is observed twice. If these observations are independent, verify that the
different samples of size 2 and their probabilities are as shown in the next column (you don’t need to do
this, just use the table). Not e that solutions are in the back of the book, so you must show work.
(a)
(b)
(c)
(d)
Find the sampling distribution of the sample mean 𝑥̅ .
Construct a probability histogram for the sampling distribution of 𝑥̅ .
What is the probability that 𝑥̅ is 4.5 or larger?
Would you expect to observe a value of 𝑥̅ equal to 4.5 or larger? Explain.
2. (5.4, page 272 ) Refer to the previous question and find 𝐸(𝑥) = 𝜇. Then use the sampling
distribution of 𝑥̅ found in the previous question to find the expected value of 𝑥̅ . Note that
𝐸(𝑥̅ ) = 𝜇.
3. (5.12, page 276) Refer to the first question (5.3).
(a) Show that 𝑥̅ is an unbiased estimator of 𝜇.
(b) Find 𝜎𝑥̅2 .
(c) Find the probability that 𝑥̅ will fall within 2𝜎𝑥̅ of 𝜇.
4. (5.13, page 276) Refer to the first question (5.3). Solutions are in the back of the book so you
must show work to explain the answers.
a) Find the sampling distribution of 𝑠 2 .
b) Find the population variance 𝜎 2 .
c) Show that 𝑠 2 is an unbiased estimator of 𝜎 2 .
d) Find the sampling distribution of the sample standard deviation s.
e) Show that s is a biased estimator of 𝜎.
5. (5.16, page 283) Suppose a random sample of 𝑛 = 25 measurements is selected from a
population with mean 𝜇 and standard deviation 𝜎. For each of the following values of 𝜇 and 𝜎,
give the values of 𝜇𝑥̅ and 𝜎𝑥̅ .
a) 𝜇 = 10, 𝜎 = 3
b) 𝜇 = 100, 𝜎 = 25
c) 𝜇 = 20, 𝜎 = 40
d) 𝜇 = 10, 𝜎 = 100
6. (5.24, page 284) Salary of a travel management professional. According to a National Business
Travel Association (NBTA) 2010 survey, the average salary of a travel management professional
is $96, 850. Assume that the standard deviation of such salaries is $30,000. Consider a random
sample of 50 travel management professionals and let 𝑥̅ represent the mean salary for the
sample.
a) What is 𝜇𝑥̅ ?
b) What is 𝜎𝑥̅ ?
c) Describe the shape of the sampling distribution of 𝑥̅ .
d) Find the z-score for the value 𝑥̅ = 89,500.
e) Find 𝑃(𝑥̅ > 89,500).
7. (5.44, page 289) Working on summer vacation. According to a poll of US adults, about 45% work
during their summer vacation. Assume that the true proportion of all US adults that work
during summer vacation is 𝑝 = 0.45. Now consider a random sample of 500 US adults.
a) What is the probability that between 40% and 50% of the sampled adults work during
summer vacation?
b) What is the probability that over 60% of the sampled adults work during summer vacation?
Good practice questions from the book (not to be handed in, answers in back of book):
5.9, 5.17, 5.21, 5.23, 5.37, 5.39, 5.43