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The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
CHAPTER 7Sampling Distributions
7.3
Sample Means
The Practice of Statistics, 5th Edition 2
7.2 Multiple choice
Page 449: 43-46
43. A
44. B
45. B
46. B
The Practice of Statistics, 5th Edition 3
Homework
38. What sample size would be required to reduce the standard deviation of
the sampling distribution to one-third the value you found in Exercise 36?
Because the standard deviation is found by dividing by ๐, using 9n for the
sample size reduces the standard deviation of the sampling distribution to one-
third of the previous value ( 9๐ = 3 ๐); we would need to sample 9(1785)
=16,065 adults.
The Practice of Statistics, 5th Edition 4
Homework
40. Harley-Davidson motorcycles make up 14% of all the motorcycles
registered in the United States. You plan to interview an SRS of 500
motorcycle owners. How likely is your sample to contain 20% or more
who own Harleys? Show your work.
We want to find the probability that more than 20% of those sampled own a
Harley with n = 500 and p = 0.14.
๐ เท๐ = ๐ = 0.14
Since it is reasonable that there are more than 10(500) = 5000 motorcycle
owners in the US, ๐ เท๐ =(0.14)(0.86)
500= 0.0155
We can use the Normal approximation because 500 0.14 = 70 โฅ 10 and
500 0.86 = 430 โฅ 10.
Step 1: ๐ ฦธ๐ โฅ 0.20 with ๐ 0.14, 0.0155
Step 2: ๐ง =0.2โ0.14
0.155= 3.87; ๐ ฦธ๐ โฅ 20 = 1 โ ๐ ๐ง โค 3.87 โ 0;
๐๐๐๐๐๐๐๐๐ ๐๐๐ค๐๐: 0.20, ๐ข๐๐๐๐: 1000, ๐: 0.14, ๐: 0.0155 = 0.00005Step 3: There is a 0.0001 probability that 20% or more of the motorcycle
owners in the sample own Harleys.
The Practice of Statistics, 5th Edition 5
Homework42. The Harvard College Alcohol Study finds that 67% of college students support
efforts to โcrack down on underage drinking.โ Does this result hold at a large local
college? To find out, college administrators survey an SRS of 100 students and
find that 62 support a crackdown on underage drinking.
a) Suppose that the proportion of all students attending this college who support
a crackdown is 67%, the same as the national proportion. What is the
probability that the proportion in an SRS of 100 students is 0.62 or less?
Show your work.
We want to find the probability that the proportion of students sampled who favor a โcrackdown on
underage drinkingโ is 0.62 or less with n = 100 and p = 0.67.
๐ เท๐ = ๐ = 0.67
Since it is reasonable that there are more than 10(100) = 1000 students at a โlarge local collegeโ,
๐ เท๐ =(0.67)(0.33)
100= 0.0470
We can use the Normal approximation because 100(0.67) = 67 โฅ 10 and 100 0.33 = 33 โฅ 10.
Step 1: ๐ ฦธ๐ โค 0.62 with ๐(0.67, 0.0470)
Step 2: ๐ง =0.62โ0.67
0.0470= โ1.06; ๐ ฦธ๐ โค 0.62 = ๐ ๐ง โค โ1.06 = 0.1446;
๐๐๐๐๐๐๐๐๐ ๐๐๐ค๐๐:โ1000, ๐ข๐๐๐๐: 0.62, ๐: 0.67, ๐: 0.0470 = 0.1437Step 3: There is a 0.1437 probability that 62% or fewer students in an SRS of 100 support a
crackdown on underage drinking.
The Practice of Statistics, 5th Edition 6
Homework42. The Harvard College Alcohol Study finds that 67% of college students support
efforts to โcrack down on underage drinking.โ Does this result hold at a large local
college? To find out, college administrators survey an SRS of 100 students and
find that 62 support a crackdown on underage drinking.
b) A writer in the collegeโs student paper says that โsupport for a crackdown is
lower at our school than nationally.โ Write a short letter to the editor
explaining why the survey does not support this conclusion.
Because the probability isnโt very small, it is
plausible that the 67% claim is correct and that
the lower than expected percentage is due to
chance alone.
Learning Objectives
After this section, you should be able to:
The Practice of Statistics, 5th Edition 7
โ FIND the mean and standard deviation of the sampling distribution
of a sample mean. CHECK the 10% condition before calculating the
standard deviation of a sample mean.
โ EXPLAIN how the shape of the sampling distribution of a sample
mean is affected by the shape of the population distribution and the
sample size.
โ If appropriate, use a Normal distribution to CALCULATE
probabilities involving sample means.
Sample Means
The Practice of Statistics, 5th Edition 8
The Sampling Distribution of
When we record quantitative variables we are interested in other
statistics such as the median or mean or standard deviation. Sample
means are among the most common statistics.
Consider the mean household earnings for samples of size 100.
Compare the population distribution on the left with the sampling
distribution on the right. What do you notice about the shape, center,
and spread of each?
x
The Practice of Statistics, 5th Edition 9
The Sampling Distribution of
When we choose many SRSs from a population, the sampling
distribution of the sample mean is centered at the population mean ยต
and is less spread out than the population distribution. Here are the
facts.
Sampling Distribution of a Sample Mean
as long as the 10% condition is satisfied: n โค (1/10)N.
x
The standard deviation of the sampling distribution of x is
s x =s
n
The mean of the sampling distribution of x is mx = m
Suppose that x is the mean of an SRS of size n drawn from a large population
with mean m and standard deviation s. Then :
Note: These facts about the mean and standard deviation of x are true
no matter what shape the population distribution has.
The Practice of Statistics, 5th Edition 10
Sampling From a Normal Population
We have described the mean and standard deviation of the sampling
distribution of the sample mean x but not its shape. That's because the
shape of the distribution of x depends on the shape of the population
distribution.
In one important case, there is a simple relationship between the two
distributions. If the population distribution is Normal, then so is the
sampling distribution of x . This is true no matter what the sample size is.
Sampling Distribution of a Sample Mean from a Normal Population
Suppose that a population is Normally distributed with mean m
and standard deviation s . Then the sampling distribution of x
has the Normal distribution with mean m and standard
deviation s / n, provided that the 10% condition is met.
The Practice of Statistics, 5th Edition 11
The Central Limit Theorem
Most population distributions are not Normal. What is the shape of the
sampling distribution of sample means when the population distribution
isnโt Normal?
It is a remarkable fact that as the sample size increases, the distribution
of sample means changes its shape: it looks less like that of the
population and more like a Normal distribution!
When the sample is large enough, the distribution of sample means is
very close to Normal, no matter what shape the population distribution
has, as long as the population has a finite standard deviation.
Draw an SRS of size n from any population with mean m and finite
standard deviation s. The central limit theorem (CLT) says that when n
is large, the sampling distribution of the sample mean x is approximately
Normal.
The Practice of Statistics, 5th Edition 12
The Central Limit Theorem
Consider the strange population distribution
from the Rice University sampling distribution
applet.
Describe the shape of the sampling
distributions as n increases. What do you
notice?
The Practice of Statistics, 5th Edition 13
The Central Limit Theorem
Normal/Large Condition for Sample Means
If the population distribution is Normal, then so is the
sampling distribution of x . This is true no matter what
the sample size n is.
If the population distribution is not Normal, the central
limit theorem tells us that the sampling distribution
of x will be approximately Normal in most cases if
n ยณ 30.
The central limit theorem allows us to use Normal probability
calculations to answer questions about sample means from many
observations even when the population distribution is not Normal.
As the previous example illustrates, even when the population
distribution is very non-Normal, the sampling distribution of the sample
mean often looks approximately Normal with sample sizes as small as
n = 25.
The Practice of Statistics, 5th Edition 14
The Sampling Distribution of
x
The Practice of Statistics, 5th Edition 15
Mean texts
Suppose that the number of texts sent during a typical day by students
at a particular high school follows a right-skewed distribution with a
mean of 45 and a standard deviation of 35.
Problem: How likely is it that a random sample of 50 students will have
sent more than a total of 2500 texts in the last day?
Let าง๐ฅ = the average number of texts sent by 50 randomly selected students
๐ าง๐ฅ= ๐ = 45We must assume that this high school has at least 10(50) = 500 students, so
๐ าง๐ฅ =๐
๐=
35
50= 4.95
Because of the Central Limit Theorem, we can use the normal distribution
because ๐ = 50 โฅ 30.Step 1: ๐ าง๐ฅ > ฮค2500
50 = ๐( าง๐ฅ > 50) with N(45, 4.95)
Step 2: ๐ง =50โ45
4.95= 1.01; ๐ าง๐ฅ > 50 = ๐ ๐ง > 1.01 = 1 โ ๐ ๐ง โค 1.01 = 1 โ
0.8438 = 0.1562; ๐ก๐๐โ: ๐๐๐๐๐๐๐๐๐ 50, 1000,45,4.95 = 0.1562Step 3: The probability that a random sample of students will have sent more
than 2500 in the last day is 0.1562.
The Practice of Statistics, 5th Edition 16
Buy me some peanuts and sample means
Problem: At the P. Nutty Peanut Company, dry-roasted, shelled
peanuts are placed in jars by a machine. The distribution of weights in
the jars is approximately Normal, with a mean of 16.1 ounces and a
standard deviation of 0.15 ounces.
(a) Without doing any calculations, explain which outcome is more
likely: randomly selecting a single jar and finding that the contents
weigh less than 16 ounces or randomly selecting 10 jars and finding
that the average contents weigh less than 16 ounces.
Because averages are less variable than individual measurements,
you would expect the sample mean of 10 jars to be closer, on
average, to the true mean of 16.1 ounces. Thus, it is more likely that a
single jar would weigh less than 16 ounces than for the average
weight of 10 jars to be less than 16 ounces.
The Practice of Statistics, 5th Edition 17
Buy me some peanuts and sample means
Problem: At the P. Nutty Peanut Company, dry-roasted, shelled
peanuts are placed in jars by a machine. The distribution of weights in
the jars is approximately Normal, with a mean of 16.1 ounces and a
standard deviation of 0.15 ounces.
(b) Find the probability that a randomly selected jar contains less than
16 ounces of peanuts.
Step 1: Let X = the weight of the contents of a randomly selected jar of
peanuts. X has the approximately N(16.1, 0.15) distribution, and we
want to find ๐(๐ < 16).
Step 2: ๐ง =16โ16.1
0.15= โ0.67;๐ ๐ < 16 = ๐ ๐ง < โ0.67 = 0.2514;
Tech:๐๐๐๐๐๐๐๐๐ โ1000, 16, 16.1, 0.15 = 0.2525Step 3: There is about a 25% chance that a single jar will contain less
than 16 ounces of peanuts.
The Practice of Statistics, 5th Edition 18
Buy me some peanuts and sample means
Problem: At the P. Nutty Peanut Company, dry-roasted, shelled
peanuts are placed in jars by a machine. The distribution of weights in
the jars is approximately Normal, with a mean of 16.1 ounces and a
standard deviation of 0.15 ounces.
(c) Find the probability that 10 randomly selected jars contain less than
16 ounces of peanuts, on average.
Let าง๐ฅ = the average weight of the contents of 10 randomly selected jars of peanuts.
๐ าง๐ฅ = ๐ = 16.1Since it is reasonable that 10 is less than 10% of all jars of peanuts produced, ๐ าง๐ฅ =๐
๐=
0.15
10= 0.047.
Since the original distribution was approximately Normal, we can use the Normal
approximation here, soโฆ
Step 1: าง๐ฅ has roughly the N(16.1, 0.047) distribution and we want to find ๐( าง๐ฅ < 16).
Step 2: ๐ง =16โ16.1
0.047= โ2.13; ๐ าง๐ฅ < 16 = P z < โ2.13 = 0.0166; Tech:
๐๐๐๐๐๐๐๐๐ โ1000, 16, 16.1, 0.047 = 0.0167.
Step 3: There is about a 1.7% chance that a random sample of 10 jars will contain
less than 16 ounces of peanuts, on average. This confirms our answer to part (a).
Section Summary
In this section, we learned how toโฆ
The Practice of Statistics, 5th Edition 19
โ FIND the mean and standard deviation of the sampling distribution of
a sample mean. CHECK the 10% condition before calculating the
standard deviation of a sample mean.
โ EXPLAIN how the shape of the sampling distribution of a sample
mean is affected by the shape of the population distribution and the
sample size.
โ If appropriate, use a Normal distribution to CALCULATE probabilities
involving sample means.
Sample Means
The Practice of Statistics, 5th Edition 20
PAGE 46250-58EVEN
Homework
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