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STAT 3011
Homework 4
(Due: February 25, 2024)
Spring 2024
Chapter covered: 7
Show work for full credit.
1. According to the American Red Cross, 7 % of people in the United States have blood type O-negative.
For a particular person, let x=1 if they have blood type o-negative and x=0 otherwise. For a random
sample of size 90,
a. State the population distribution (that is, the probability distribution of X containing values of
X and probabilities)
b. State the data distribution if 9 of the 90 people sampled have blood type O-negative. the data
distribution contains the values of X and probabilities/proportions
c. Find the mean of the sampling distribution of the sample proportion of people with blood type
O-negative among the 90 people.
d. Find the standard deviation of the sampling distribution of the sample proportion of people with
blood type O-negative among the 90 people.
e. Explain what the standard deviation of the sampling distribution of the sample proportion with
blood type O-negative among the 90 people in part d describes
2. According to Mediamark Research, Inc., 84 % of all U.S. households own a cellular phone (one or
more).
(a) Find the standard deviation for the sampling distribution of the sample proportion with (i) n=100
and (ii) n=1000
1
STAT 3011
Homework 4
(Due: February 25, 2024)
Spring 2024
(b) Explain why the sample proportion would be very likely to fall (i) between 0.73 and 0.95 when
n=100, and (ii) between 0.80 and 0.88 when n=1000 (Hint: Recall that for an approximately
normal distribution nearly the entire distribution is within 3 standard deviations of
the mean)
(c) Explain how the results in part b indicate that the sample proportion tends to estimate the
population proportion more precisely/accurately when the sample size is large.
3. According to the Cable and Telecommunications Association for Marketing, 30 % of cable television
subscribers owned a high-definition television (HDTV) in 2007.
(a) Find the mean and the standard deviation of the sampling distribution for the proportion of
cable television subscribers owning a high-definition television in 2007 in a random sample of 200
television subscribers.
(b) Is it reasonable to assume a normal shape for this sampling distribution? Explain
(c) How likely is it that we get fewer than 35 % of cable television subscribers own an HDTV in the
given random sample of 200 cable television subscribers?
2
STAT 3011
Homework 4
(Due: February 25, 2024)
Spring 2024
(d) If instead n=1000, how likely is it then that fewer than 35 % of the cable television subscribers
own an HDTV?
4. The Food and Drug Administration sets Food Defects Action Levels (FDALs) for some of the various
foreign substances that inevitably end up in the food we can eat and the liquids we drink. For example,
the FDAL for insect fifth in peanut butter is 3 insect fragments (larvae, eggs, body parts, and so on)
per 100 grams. A random sample of 40 ten-gram portions of peanut butter is obtained and results in
a sample mean of x̄ = 3.6 insect fragments per ten-gram portion and a sample standard deviation of
s=1.2
(a) What is the mean and the standard deviation of the data distribution?
(b) Find the mean and the standard
deviation of the sampling distribution of the sample mean
√
assuming that µ = 3 and σ = 3
(c) Is the sample mean of 2 insect fragments unusually small? Find its z-score and comment.
(d) What is the probability that a sample of 50 ten-gram portions results in a mean of at least 3.2
insect fragments? Is this unusual
3
STAT 3011
Homework 4
(Due: February 25, 2024)
Spring 2024
5. According to ATMDepot.com, the mean ATM withdrawal is $ 60. Assume that the standard deviation
for withdrawals is $ 35.
(a) Describe the center and the variability of the population distribution. What shape does it probably
have? most people withdraw a small amount of money
(b) If a random sample of 50 ATM withdrawals is obtained, describe the sampling distribution of the
sample mean.
(c) Explain why it would not be unusual to obtain a withdrawal amount of $30 but highly unusual
to observe a sample mean withdrawal amount of less than $ 30 if a random sample of 50 ATW
withdrawals is obtained. Use probabilities to answer the part involving the sample mean
and check your answer using the values of z scores
(d) Determine the probability of obtaining a sample mean withdrawal amount more than $75
(e) Determine the probability of obtaining a sample mean withdrawal amount between $ 55 and $75
4
STAT 3011
Homework 4
(Due: February 25, 2024)
Spring 2024
6. In the following exercises indicate whether the Central Limit Theorem applies so that the sample
proportions follow a normal distribution.
a. n=500 and p=0.2
b. n=20 and p=0.5
c. n=30 and p=0.2
d. n=100 and p=0.8
7. According to a study conducted by the Gallup organization, the proportion of Americans who were
afraid to fly in 2006 was 0.1
5
STAT 3011
Homework 4
(Due: February 25, 2024)
Spring 2024
(a) A random sample of 1100 Americans results in 121 indicating that they are afraid to fly. What is
the observed sample proportion and what is the distribution of the sample proportion of Americans
that are afraid to fly?
(b) A random sample of 2000 Americans results in 121 indicating that they are afraid to fly. What is
the observed sample proportion and what is the distribution of the sample proportion of Americans
that are afraid to fly?
(c) What effect does increasing sample size have on the standard deviation (standard error) of the
distribution of the sample proportions? Explain.
(d) If the sample size increases by a factor of 16, what happens to the standard deviation of p̂? be
precise
Problem 8 R Problems
Use the Getting To Know You Survey dataset to answer the following questions.
We are interested in answering questions about what proportion of people in MN prefer the Summer
season.
Getting2NoU
Purchase answer to see full
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