Download Sampling Distributions

SAMPLING DISTRIBUTIONS
Chapter 8
DISTRIBUTIONS OF THE SAMPLE MEAN
Lesson 8.1
SAMPLING STATISTICS
Since statistics are actually random variables
associated with a given sample, they will vary from
sample to sample. Therefore, they have
probabilities distributions associated with them. This
will allows us to find probabilities associated with
the sample.
i.e. what is the probability that the mean of the
population matches the mean of your sample.
SAMPLING DISTRIBUTIONS ABOUT THE MEAN
1. Obtain a simple random sample of size n.
2. Compute the sample mean.
3. Repeat steps 1 and 2 until all simple random
samples have been obtained from the population.
EXAMPLE
The weights of pennies minted after 1982 are
approximately normally distributed with mean 2.46 grams
and standard deviation 0.02 grams.
Approximate the sampling distribution of the sample mean
by obtaining 200 simple random samples of size n = 5
from this population.
The data on the following slide represent the sample
means for the 200 simple random samples of size n = 5.
For example, the first sample of n = 5 had the following
data:
2.493
Note:
2.466 2.473 2.492 2.471
x = 2.479 for this sample
EFFECT OF SAMPLE SIZE
Repeat Experiment using sample size of n = 20
The mean of the 200 sample means for n =20 is
still 2.46, but the standard deviation is now
0.0045 (0.0086 for n = 5).
As expected, there is less variability in the
distribution of the sample mean with n =20 than
with n =5.
MEAN AND STANDARD DEVIATION OF 𝑥
Suppose a simple random sample of size n is drawn
from a large population with mean µ and standard
deviation 𝜎. The sampling distribution of 𝑥 will have:
𝜇𝑥 = 𝜇
𝜎𝑥 =
𝜎
𝑛
EXAMPLE
Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4
minutes with a standard deviation of 3.2 minutes
1. If a random sample of n = 35 oil changes I selected describe the sampling
distribution of the sample mean.
2. If a random sample of n = 35 oil changes is selected, what is the probability that the
mean oil change time is less than 11 minutes.
#21: THE LENGTH OF HUMAN PREGNANCIES IS APPROXIMATELY NORMALLY
DISTRIBUTED WITH MAN 266 DAYS AND STANDARD DEVIATION OF 16 DAYS
a) What is the probability a randomly selected pregnancy lasts less than
260 days.
b) Suppose a random sample of 20 pregnancies is obtained. Describe the
sampling distribution of the sample mean length of human pregnancies.
c) What is the probability that a random sample of 20 pregnancies has a
mean gestation period of 260 days or less?
d) What is the probability that a random sample of 50 pregnancies has a
mean gestation period of 260 days or less?
e) What might you conclude if a random sample of 50 pregnancies
resulted in a mean gestation period of 260 days or less?
DISTRIBUTION OF SAMPLE
PROPORTIONS
Lesson 8.2
POINT ESTIMATE OF A POPULATION PROPORTION
Number of individuals of a sample with a certain
characteristic
Number of people whose favorite color is red
Number of roses in the greenhouse with yellow flowers
Sample Proportion (p-hat)  𝑝 =
𝑥
𝑛
EXAMPLE
In a Quinnipiac University Poll conducted in May of 2008,
1,745 registered voters nationwide were asked whether
they approved of the way George W. Bush is handling the
economy. 349 responded “yes”. Obtain a point estimate
for the proportion of registered voters who approve of the
way George W. Bush is handling the economy.
Using Simulation to Describe the
Distribution of the Sample Proportion
According to a Time poll conducted in June
of 2008, 42% of registered voters
believed that gay and lesbian couples
should be allowed to marry.
Describe the sampling distribution of the
sample proportion for samples of size
n=10, 50, 100.
8-17
8-18
8-19
8-20
Key Points from Example 2
Shape: As the size of the sample, n, increases, the shape
of the sampling distribution of the sample proportion
becomes approximately normal.
Center: The mean of the sampling distribution of the
sample proportion equals the population proportion, p.
Spread: The standard deviation of the sampling
distribution of the sample proportion decreases as the
sample size, n, increases.
8-21
SAMPLING DISTRIBUTION CHARACTERISTICS
1. Shape is approximately normal provided
𝑛𝑝 1 − 𝑝 ≥ 10
2. Mean of the sampling distribution is 𝜇𝑝 = 𝑝
3. Standard deviation of sampling distribution is
𝜎𝑝 =
𝑝(1−𝑝)
𝑛
4. Values independent ….. Verify n is no more than
5% of the population
According to a Time poll conducted in June of 2008, 42% of
registered voters believed that gay and lesbian couples should be
allowed to marry. Suppose that we obtain a simple random sample of
50 voters and determine which believe that gay and lesbian couples
should be allowed to marry. Describe the sampling distribution of
the sample proportion for registered voters who believe that gay and
lesbian couples should be allowed to marry.
COMPUTE PROBABILITIES OF SAMPLE PROPORTIONS
According to the Centers for Disease Control and Prevention,
18.8% of school-aged children, aged 6-11 years, were
overweight in 2004.
(a) In a random sample of 90 school-aged children, aged 611 years, what is the probability that at least 19% are
overweight?
(b) Suppose a random sample of 90 school-aged children,
aged 6-11 years, results in 24 overweight children. What
might you conclude?
#17: ACCORDING TO A USA TODAY SNAPSHOT, 26% OF ADULTS DO NOT HAVE
ANY CREDIT CARDS. A SIMPLE RANDOM SAMPLE OF 500 ADULTS IS OBTAINED.
8-25