Download tps5e_Ch7_3

CHAPTER 7
Sampling
Distributions
7.3
Sample Means
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Sample Means
Learning Objectives
After this section, you should be able to:
 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.
 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.
 If appropriate, use a Normal distribution to CALCULATE
probabilities involving sample means.
The Practice of Statistics, 5th Edition
2
The Sampling Distribution of x
The Practice of Statistics, 5th Edition
3
Activity: Penny for Your Thoughts
The Practice of Statistics, 5th Edition
4
Activity: Penny for Your Thoughts
5. Place a sticky for each of the five ages of your pennies on the ages
dotplot.
6. Does this distribution of ages surprise you?
7. Now place a sticky for the mean age on the mean age dotplot.
8. What differences do you notice between the dotplots?
The Practice of Statistics, 5th Edition
5
Note:
1. The histogram of the ages of the individual pennies is the graph of
the population.
2. The histogram of the means of the five ages is the graph of the
sampling distribution of the sample mean for samples of size 5.
3. These distributions WERE NOT THE SAME. Sampling distributions
are DIFFERENT DISTRIBUTIONS than population distributions.
The three facts listed above are the three facts that make up the
Central Limit Theorem. But before we discuss that in more detail, let’s
look at a what a larger class would have seen if they did the same
activity.
The Practice of Statistics, 5th Edition
6
Note:
Suppose my class had 350 students in it, and each of them did the
same activity. So there are 5 x 350 = 1750 pennies. And there are 350
means. We find the same skewed shape for the ages of pennies that
we saw in the smaller class. But now there are enough means that we
can see more clearly the shape of the distribution of means.
The Practice of Statistics, 5th Edition
7
Note:
Now we can notice several things about this graph of means.
a. It is not nearly as spread out as the distribution of the individual ages.
(Look carefully at the scale along the horizontal axis to see this.)
–
b.
c.
That means the standard deviation is smaller.
The center of it is about the same as the center of the distribution of the
individual ages.
It is still skewed, but not nearly as skewed as the distribution of the
individual ages. With this many observations, it is clear that it looks more
normally distributed than the distribution of individual ages.
The Practice of Statistics, 5th Edition
8
The Sampling Distribution of x
This example illustrates an important fact that we
will make precise in this section: averages are
less variable than individual observations.
The Practice of Statistics, 5th Edition
9
The Sampling Distribution of
x
When we choose many SRSs from a population, the sampling
distribution of the sample mean is centered at the population mean µ
and is less spread out than the population distribution. Here are the
facts.
Sampling Distribution of a Sample Mean
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 :
The mean of the sampling distribution of x is mx = m
The standard deviation of the sampling distribution of x is
sx =
s
n
as long as the 10% condition is satisfied: n ≤ (1/10)N.
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
The Practice of Statistics, 5th Edition
11
Ex: This Wine Stinks
The Practice of Statistics, 5th Edition
12
The Practice of Statistics, 5th Edition
13
Sampling From a Normal Population
The Practice of Statistics, 5th Edition
14
In this Activity, you’ll use Professor Lane’s applet to explore the shape
of the sampling distribution when the population is Normally distributed.
1. There are choices for the population distribution: Normal, uniform,
skewed, and custom. The default is Normal. Click the “Animated”
button. What happens? Click the button several more times.
2. What do the black boxes represent? What is the blue square that
drops down onto the plot below? What does the red horizontal
band under the population histogram tell us?
3. Click on “Clear lower 3” to start clean. Then click on the “10,000”
button under “Sample:” so you simulate taking 10,000 SRSs of
size n = 5 from the population.
4. Does the approximate sampling distribution (blue bars) have a
recognizable shape? Click the box next to “Fit normal.”
The Practice of Statistics, 5th Edition
15
The Practice of Statistics, 5th Edition
16
Sampling From a Normal Population
Sampling Distribution of a Sample Mean from a Normal Population
Suppose that a population is Normally distribute d with mean 
and standard deviation  . Then the sampling distributi on of x
has the Normal distributi on with mean  and standard
deviation

n
, provided that the 10% condition is met.
The Practice of Statistics, 5th Edition
17
Ex: Young Women’s Heights
PROBLEM: The height of young women follows a Normal distribution
with mean μ = 64.5 inches and standard deviation σ = 2.5 inches.
a. Find the probability that a randomly selected young woman is taller
than 66.5 inches. Show your work.
b. Find the probability that the mean height of an SRS of 10 young
women exceeds 66.5 inches. Show your work.
The Practice of Statistics, 5th Edition
18
On Your Own:
The Practice of Statistics, 5th Edition
19
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
isn’t Normal?
Let’s find out with the same Applet.
The Practice of Statistics, 5th Edition
20
1. Select “Skewed” population. Set the bottom two graphs to display
the mean—one for samples of size 2 and the other for samples of
size 5. Click the Animated button a few times to be sure you see
what’s happening. Then “Clear lower 3” and take 10,000 SRSs.
Describe what you see.
2. Change the sample sizes to n = 10 and n = 16 and repeat Step
1. What do you notice?
3. Now change the sample sizes to n = 20 and n = 25 and take
10,000 more samples. Did this confirm what you saw in Step 2?
The Practice of Statistics, 5th Edition
21
The Central Limit Theorem
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.
This famous fact of probability theory is called the central limit
theorem (sometimes abbreviated as CLT).
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
22
CAUTION:
The Practice of Statistics, 5th Edition
23
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
24
The Central Limit Theorem
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.
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 ³ 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.
The Practice of Statistics, 5th Edition
25
Ex: Servicing Air Conditioners
The Practice of Statistics, 5th Edition
26
The Sampling Distribution of
The Practice of Statistics, 5th Edition
x
27
Sample Means
Section Summary
In this section, we learned how to…
 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.
 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.
 If appropriate, use a Normal distribution to CALCULATE probabilities
involving sample means.
The Practice of Statistics, 5th Edition
28