Sampling Sampling Methods and the Central Limit Theorem.pdf

Sampling, Sampling Methods, and
the Central Limit Theorem
Chapter 8
Copyright © 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the
prior written consent of McGraw-Hill Education.
8-1

Learning Objectives
Copyright © 2022 McGraw-Hill Education.All rights reserved. No reproduction or distribution without
the prior written consent of McGraw-Hill Education.
LO8-1 Explain why populations are sampled and
describe four methods to sample a
population
LO8-2 Define sampling error
LO8-3 Explain the sampling distribution of the
sample mean.
LO8-4 Explain how the central limit theorem
applies to the sampling distribution of the
sample mean
LO8-5 Apply the central limit theorem to
calculate probabilities
8-2

Reasons for Sampling a Population
1. To contact the whole populations would be time
consuming.
2. The cost of studying all the items in a populations may
be prohibitive.
3. The physical impossibility of checking all items in the
population.
4. The destructive nature of some tests.
8-3

Probability Sampling Methods
 Simple random sample – a sample selected so that each item
or person in the population has the same probability or
chance of being selected
 Systematic random sample – a random starting point is
selected, and then every kth member of the population is
selected
 Stratified random sample – a population is divided into
subgroups, called strata, and a sample is selected from each
stratum
 Clustered sampling – a population is divided into clusters using
naturally occurring geographic or other boundaries.Then,
clusters are randomly selected and a sample is collected by
randomly selecting from each cluster
8-4

Simple Random Sampling
 The most widely used method of sampling is a simple
random sample
8-5

Using a Table of Random Numbers
Suppose the population of interest is the 750 Major League Baseball players on the
active rosters of the 30 teams at the end of the last season.A committee of 10 players
is to be formed to study the issue of concussions.To make sure every player has an
equal chance of being selected, use a table of random numbers.
1. Prepare of list of all the players and number them 1 through 750
2. Randomly pick a starting place in the random number table
3. Select 10 three-digit numbers between 1 and 750
8-6

Systematic Random Sampling
 If you do not have a list of the entire population to begin
with, you can use the systematic random sample
 Example
 Stood’s Grocery Store wants to study the length of time
customers spend in their store
 Randomly select the days of the week, the times, and the
starting point of the study, then systematically select the
customers and measure the time each spends in the store
8-7

Stratified Random Sampling
 When the population can be divided into groups based
on some characteristic, use stratified random sampling
 Example
 A study of 50 of the 352 largest US firms’ ad spending
 Begin by identifying the strata, then use random sampling
within each group based on relative frequencies to collect the
sample
8-8

Cluster Sampling
 Cluster sampling is a common type of sampling, used to
reduce the cost of sampling over large geographic areas
 Example
 Suppose we wish to sample residents
of the 12 counties in the greater Chicago
area about their views on government policy.
 Randomly select 3 counties and then select a
random sample of the residents in each
of the 3 counties.
8-9

Sampling Error
 It is unlikely the mean of a sample will be exactly equal to
the mean of the population
 Example
 The Foxtrot Inn’s number of rooms rented in June.The mean
number of rooms rented, μ, is 3.13
 Taking three random samples of size
5, we find sample means, ത
x, of 3.80,
3.40 and 1.80.The sampling error is
the difference between each ത
x and μ
8-10

Sampling Distribution of the Sample Mean
 How can we determine how accurate the sample mean
is?
1. The mean of the sample means is exactly equal to the
population mean.
2. The dispersion of the sampling distribution of the sample
mean is narrower than the population distribution.
3. The sampling distribution of the sample mean tends to
become bell shaped and to approximate the normal
probability distribution
8-11

Sampling Distribution Example
Tartus Industries has seven production employees (the population).
The hourly earnings of each employee is given in the table.
1. What is the population mean?
2. What is the sampling distribution of the sample mean for
samples of size 2?
3. What is the mean of the sampling distribution?
4. What observations can be made about the population and the
sampling distribution?
8-12

Sampling Distribution Example (2 of 5)
The hourly earnings of each employee is given in the table.
1. What is the population mean?
μ =
Σx
N
= $15.43
8-13

2. What is the sampling distribution of the sample mean for
samples of size 2?
8-14

3.What is the mean of the sampling distribution?
8-15

 The mean of the distribution of the sample mean ($15.43) is
equal to the mean of the population, μ = μഥ
x
 The spread in the distribution of the sample mean is less than
the spread in the population values
 The shapes of the population and sample distributions are
different
4.What observations can be made about the population
and the sampling distribution?
8-16

Central Limit Theorem
 If the population follows a normal probability distribution, then
for any sample size the sampling distribution of the sample
mean will also be normal
 If the population distribution is symmetrical, you will see the
normal shape of the distribution of the sample mean emerge
with samples as small as 10
 If the distribution is skewed or has thick tails, it may require
samples of 30 or more to observe the normality feature
8-17

Central Limit Theorem Results
8-18

Central Limit Theory Conclusions
 The mean of the distribution of sample means will be
exactly equal to the population mean, if we select all
possible samples of same size from the population
μ = μഥ
x
 The standard deviation of the sampling distribution of the
sample mean is also called the standard error of the mean
 There will be less dispersion in the sampling distribution
of the sample mean, σ/ n, than in the population σ
8-19

Sampling Distribution of the Sample Mean
 If the population follows a normal distribution, the
sampling distribution of the sample mean will also follow
the normal distribution for samples of any size
 If the population is not normally distributed, the sampling
distribution of the sample mean will approach a normal
distribution when the sample size is at least 30
 Assume the population standard deviation is known
 To determine the probability that a sample mean falls in a
particular region, use the following formula:
8-20

Using the Sampling Distribution Example
The Quality Assurance Dept. for Cola, Inc. maintains records regarding the
amount of cola in its jumbo bottle.The actual amount of cola in each bottle
varies a small amount from one bottle to another. Records indicate the
amounts of cola follow the normal distribution, the mean amount of cola in the
bottles is 31.2 ounces, and the standard deviation is 0.4 ounces.At 8 a.m. today,
the quality technician randomly selected 16 bottles from the filling line.The
mean amount was 31.38 ounces. Is this an unlikely result? Is it a likely the
process is putting too much soda in the bottle?
z =
ഥ
x − μ
σ/ n
=
31.38 −31.20
0.4/ 16
= 1.80
We conclude that it is unlikely;
there is less than a 4% chance.
The process is putting too much
soda in the bottles.
8-21

Question 1
8-23
The following is a list of 24 Marco’s Pizza stores in Lucas County.
The stores are identified by numbering them 00 through 23.Also
noted is whether the store is corporate owned (C) or manager
owned (M).A sample of four locations is to be selected and
inspected for customer convenience, safety, cleanliness, and other
features.
LO8-1

Question 1 (continued)
8-24
a. The random numbers selected are 08, 18, 11, 54, 02, 41, and 54.Which
stores are selected?
b. Using a random number table (Appendix B.4) or statistical software,
select your own sample of locations.
c. Using systematic random sampling, every seventh location is selected
starting with the third store in the list.Which locations will be included
in the sample?
d. Using stratified random sampling, select three locations.Two should be
corporate owned and one should be manager owned.
LO8-1

Question 1 (continued)
8-25
LO8-1
 a. 303 Louisiana,5155 S. Main, 3501 Monroe, 2652W.
Central
 b. Answers will vary.
 c. 630 Dixie Hwy, 835 S. McCord Rd, 4624Woodville
Rd
 d. Answers will vary.

Question 7
8-26
A population consists of the following five values: 12, 12, 14,
15, and 20.
a. List all samples of size 3, and compute the mean of each
sample.
b. Compute the mean of the distribution of sample means
and the population mean. Compare the two values.
c. Compare the dispersion in the population with that of
the sample means.
LO8-2

Question 7
8-27
LO8-2
a. Sample Values Sum Mean
1 12, 12, 14 38 12.66
2 12, 12, 15 39 13.00
3 12, 12, 20 44 14.66
4 14, 15, 20 49 16.33
5 12, 14, 15 41 13.66
6 12, 14, 15 41 13.66
7 12, 15, 20 47 15.66
8 12, 15, 20 47 15.66
9 12, 14, 20 46 15.33
10 12, 14, 20 46 15.33
b. μx = (12.66 + . . . + 15.33 + 15.33)/10 = 14.6
μ = (12 + 12 + 14 + 15 + 20)/5 = 14.6
c. The dispersion of the population is greater than that of the sample means.The
sample means vary from 12.66 to 16.33, whereas the population varies from 12 to
20.

Question 11
8-28
Appendix B.4 is a table of random numbers that are
uniformly distributed. Hence, each digit from 0 to 9 has the
same likelihood of occurrence.
a. Draw a graph showing the population distribution of random numbers.
What is the population mean?
b. Following are the first 10 rows of five digits from the table of random
numbers in Appendix B.4.Assume that these are 10 random samples of
five values each. Determine the mean
of each sample and plot the means on
a chart similar to Chart 8–4. Compare
the mean of the sampling distribution
of the sample mean with the
population mean.
LO8-3

Question 11
8-29
LO8-3
a. μ = (0 + 1 + . . . + 9) / 10 = 4.5
b.Sample Sum x
1 11 2.2
2 31 6.2
3 21 4.2
4 24 4.8
5 21 4.2
6 20 4.0
7 23 4.6
8 29 5.8
9 35 7.0
10 27 5.4
The mean of the 10 sample means is 4.84, which is close to the population mean of
4.5.The sample means range from 2.2 to 7.0, whereas the population values range
from 0 to 9. From the above graph, the sample means tend to cluster between 4 and 5.

Question 15
8-30
 A normal population has a mean of $60 and standard deviation of
$12.You select random samples of nine.
a. Apply the central limit theorem to describe the sampling
distribution of the sample mean with n = 9.With the small sample
size, what condition is necessary to apply the central limit
theorem?
b. What is the standard error of the sampling distribution of sample
means?
c. What is the probability that a sample mean is greater than $63?
d. What is the probability that a sample mean is less than $56?
e. What is the probability that a sample mean is between $56 and
$63?
f. What is the probability that the sampling error (x¯ − μ) would be
$9 or more? That is, what is the probability that the estimate of
the population mean is less than $51 or more than $69?
LO8-5

Question 15
8-31
LO8-5
15. a. The central limit theorem indicates that the sampling distribution has a
population mean equal to $60 and a standard error of 12/ 9 = 4. With the small sample
size of 9, the application of the central limit theorem requires that the population is
normally distributed.
b. Standard error of 12/ 9 = 4
c. the probability is 0.2266, found by 0.5000 − 0.2734.
d. the probability is 0.1587, found by 0.5000 − 0.3413
e. the probability is 0.6147, found by 0.3413 + 0.2734.
f. the probability is .5000 − .4878 = .0122 that a sample mean is more
than 2.25 standard errors from the population mean. Then, multiply by 2. Final answer
is 2(.0122) = .0244

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