Chapter 9 Section Review
If our sample is an SRS of size n, then the following statements describe the sampling model for :
The shape is Approximately Normal.
Assumption: Sample Size Is Sufficiently Large
Condition:
np
³
10
Chapter 9 Section 2 Review
If our sample is an SRS of size n, then the following statements describe the sampling model for :
The mean is exactly p.
The standard deviation is .
Assumption: Sample Size Is Sufficiently Large
Condition: Population is at least 10 times as large as the sample OR Population > 10n
p
(1
-
p
)
Chapter 9 Section 2 Review
Where did those formulas come from?
Suppose the sample size is n and the actual population is p. We learned in Section 8.1, that if X is a binomial random variable, that the mean and standard deviation of the sampling distribution of X are given by
AND .
Here, however, we are interested in the proportion rather than in the number of successes. We remember (hopefully) that when we multiply or divide every element by a constant, we multiply or
divide both the mean and standard deviation by the same constant. In this case, to change the number of successes to proportion of successes, we divide by n:
AND .
m
x=
np
s
x=
np
(1
-
p
)
m
pˆ=
np
n
=
p
s pˆ =np(1- p)
n =
p(1- p)
Example #1: Opinion polls in 2002 showed that about 70% of the population had a favorable opinion of
President Bush. That same year, a simple random sample of 600 adults living in the San Francisco Bay area found only 65% that had a favorable opinion of President Bush. What is the probability of getting a
rating of 65% or less in a random sample of this size if the true population proportion was .70?
Example #2: Trina Forest fails to study for her statistics final. The final has 100 multiple choice
Example #3: It is estimated that 48% of all motorists use their seat belts. If a police officer observes 400 cars go by in an hour, what is the probability that the proportion of drivers wearing seat belts is between 45% and 55%?
Chapter 9 Section 3
?
Sample Means
If we have categorical data, then we must use Sample Proportion to construct a sampling model.
Example #1: Suppose we want to know how many seniors in Maryland plan to attend college. We want to know how many seniors would answer, “YES” to the question, “Do you plan to attend college?” These responses are categorical.
So p (our parameter) is the proportion of all seniors Maryland
who plan to attend college. Let (our statistic) be the proportion of
Maryland students in an SRS of size 100 who plan to attend college. To
calculate the value of , we divide the number of “Yes” responses in
our sample by the total number of students in the sample.
ˆ
p
Chapter 9 Section 3
?
If I graph the values of for all possible samples of size 100, then I have constructed a sampling
model. What will the sampling model look like?
It will be Approximately Normal. In fact, the larger my sample size, the closer it will be to a Normal
model. It can never be perfectly normal, because our data is discrete, and normal distributions are
Continuous.
Chapter 9 Section 3
?
So how large is large enough to ensure that the sampling model is close to normal?
* Both np and n(1-p) should be at least 10 in order for normal approximations to be useful.
* The mean of the sampling model will Equal the true population proportion.
* The standard deviation (if the population is at least 10 times as large as the sample) will be.
If, on the other hand, we have quantitative data, then we can use Sample Means to construct a sampling model.
p(1- p)
Example #2: Suppose I randomly select 100 seniors in Maryland and record each one’s GPA. I am
Chapter 9 Section 3
?
These 100 seniors make up one possible SAMPLE.
The sample mean ( _____ ) is 2.5470 and the sample standard deviation ( _____ ) is 0.7150.
So _____ (our parameter) is the true mean GPA of a senior in Maryland. And _____ (our statistic) is the mean GPA of a senior in Maryland in an SRS of size 100.
To calculate the value of _____, we find the mean of our sample ( _____ ).
x
x
x
m
?
If we pick different samples, then the value of our statistic _____ changes:
If I graph the values of _____ for all possible samples of size 100, then I have constructed a Sampling Distribution of sample means. What will the sampling model look like?
ˆ
p
ˆ
p1 = x1 = 2.5470 ˆ
p2 = x2 = 2.4943 ˆ
p3 = x3 = 2.6223 ˆ
p4 = x4 = 2.5289 ˆ
p5 = x5 = 2.4037
ˆ
p6 = x6 = 2.3962 ˆ
p7 = x7 = 2.5019 ˆ
p8 = x8 = 2.5621 ˆ
p9 = x9 = 2.6083 ˆ
p10 = x10 = 2.5667
ˆ
Chapter 9 Section 3
?
Remember that each value is a mean. Means are Less
Variable than individual observations because if we are
looking at only means, then we do not see any extreme
values. We only see average values. We will not see GPA’s that are very low or very high, just average GPA’s.
(Averages are also more normal than individual observations!)
The larger the sample size, the less variation we will see in the values of . The standard deviation Decreases as the sample size Increases.
ˆ
p
ˆ
Chapter 9 Section 3
What will the sampling model look like?
* If the sample size is large, it will be Approximately Normal. It can never be perfectly normal, because our data is discrete, and normal distributions are
Continuous. (Does this sound familiar?)
* The mean of the sampling model will equal the true population mean .
* The standard deviation will be _________ (if the population is at least 10 times as large as the sample).
s
Example #4: The average life expectancy of a light bulb is normally distributed with a mean of 4000
Chapter 9 Section 3
Sampling Distribution of a Sample Mean
Draw an SRS of size n from a population that has the normal distribution with a mean and a standard deviation . Then the sample mean has a normal distribution N with a mean and standard deviation of
s
n
m, s
n
æ è
ç ö
ø
÷
m
,
m
,
Central Limit Theorem (CLT)
The CLT is arguably the most important theorem in statistics. The branch of statistics known as
inferential statistics relies heavily on this theorem. When you make inferences about a population mean based on information from a sample, you must know the relationship between the population from which your sample is taken and the sampling distribution of sample means. This theorem describes that
