Transcript AP Statistics Chapter 18 Part 1

AP Statistics Chapter 18
“Sampling Distributions for
Sample Proportions and Sample
Means”
What are Sampling Distributions?
 From a population, we draw a sample.
 From the sample, we collect statistics.
 From the statistics, we infer information about the
population.
 To understand the relationship between samples
and the population, we look at the sampling
distributions – the distribution of values of a
statistic taken from all possible samples of a given
size
The Distribution of Sample
Proportions
 If a simple random sample (SRS) of size n is
drawn from a large population with a
proportion p, the sampling distribution of the
sample proportion is approximately normal
with a mean of p and a standard deviation of
Sample Proportions
Mean = p
Standard Deviation =
pq
n
pq
n
Example 1
 Based on past experience, a bank believes that 7% of the
people who receive loans will not make payments on time. The
bank has recently approved 200 loans. What are the mean and
standard deviation of the proportion of clients in this group who
may not make timely payments? Draw the normal model. Label
up to three standard deviations. What is the probability that over
10% of these clients will not make timely payments?
Mean = 7%
7%
5.2%
3.4%
1.6%
8.8%
10.6%
12.4%
STD =
pq
7%  93%

 1.8%
n
200
P(over 10%):
z-score = x  mean  10%  7%  1.667
STD
1.8%
normalcdf (1.667, 99) = 4.8%
Example 2
 Public health statistics indicate that 26.4% of American adults
smoke cigarettes. Using the normal model, draw the sampling
distribution model for the proportion of smokers among a
randomly selected group of 50 adults. (Label your picture up to
three standard deviations.) What is the probability that over 33%
of this group will smoke?
Mean = 26.4%
26.4%
20.2% 32.6%
14%
38.8%
7.8%
45%
STD =
pq
26.4%  73.6%

 6.2%
n
50
P(over 33%):
z-score = x  mean  33%  26.4%  1.0645
STD
6.2%
normalcdf (1.0645, 99) = 14.4%
Conditions and Assumptions
 If these conditions are not met for your sample,
then the normal model should not be used to
approximate the sampling distribution of the
sample proportion:
 Must be an independent random sample
 The population must be at least 10 times the
sample size
 np > 10 and nq > 10 (a.k.a. minimum of 10
successes and 10 failures in the sample)
The Distribution of Sample Means
 If a simple random sample (SRS) of size n is
drawn from a large population with a mean μ,
the sampling distribution of the sample mean
is approximately normal with a mean of μ and
a standard deviation of 
n
Sample Means
Mean = μ
Standard Deviation = 
n
Example 3
 Assume that the distribution of human pregnancies can be
described by a Normal model with mean 266 days and standard
deviation 16 days. Draw a normal model for the distribution of
the mean length of the pregnancies among a sample of 60
pregnant women at a certain obstetrician’s office. Label your
picture up to three standard deviations. What is the probability
that the mean duration of these pregnancies will be less than
260 days?
Mean = 266 days
266
263.93 268.07
261.86
270.14
259.79
272.21
STD =
std
16

 2.07
n
60
P(less than 260 days):
z-score = x  mean  260  266  2.899
STD
2.07
normalcdf (-99, -2.899) = 0.19%
Conditions and Assumptions
 If these conditions are not met for your sample,
then the normal model should not be used to
approximate the sampling distribution of the
sample mean:
 Must be an independent random sample
 The population must be at least 10 times the
sample size
 The sample size n should be large (more than
forty)