Download Chap 11 Lesson 3

Chapter 11: The Sampling Distribution of
x
Lesson 3
Yesterday we saw in the homework that when you repeatedly take samples
from a population and find their means, and then calculate the overall mean and
standard deviation of those means, that the mean of the samples almost equals
the mean of the population and the standard deviation of the samples is much
smaller than that of the population. We can now generalize these concepts into
rules about sampling distributions:
Rules for Sampling Distributions of x
Suppose that x is the mean of an SRS of size n drawn from a large population
with mean µ and standard deviation σ. The sampling distribution of x has:
1. mean:  x  
2. standard deviation: x 

n
 these rules are true for any population
 the mean of the statistics( x ’s) always equals the mean of the population µ
= this means that the sampling distribution of x is centered at µ,
although some x ’s are higher than µ and some are lower than µ
(However, there is never a trend toward one direction or the other
which means it has no bias in its estimate of µ.)
= x is called an unbiased estimator of µ
o The spread of the distribution of x is always a smaller
value than σ. ( 

)
n
o averages are less variable than individual observations
o also, the results of larger samples are less variable than
results of smaller samples (but means still equal)
 shape of a distribution depends on shape of the population
= if population appears Normal, than so is the sampling distribution of x :
individual observations: N(µ, σ)
sample mean x of SRS size n: N(x , X ) = N(µ,

)
n
Chapter 11: The Sampling Distribution of
x
Lesson 3
Yesterday we saw in the homework that when you repeatedly take samples
from a population and find their means, and then calculate the overall mean and
standard deviation of those means, that the mean of the samples almost equals
the mean of the population and the standard deviation of the samples is much
smaller than that of the population. We can now generalize these concepts into
rules about sampling distributions:
Rules for Sampling Distributions of x
Suppose that x is the mean of an SRS of size n drawn from a large population
with mean µ and standard deviation σ. The sampling distribution of x has:
1. mean:
2. standard deviation:
 these rules are true for ______________________
 the mean of the statistics( x ’s) __________________________________
____________________________
= this means that the sampling distribution of x is _____________
__________, although some x ’s are higher than µ and some are
lower than µ (However, there is never a trend toward one direction
or the other which means it has no bias in its estimate of µ.)
= x is called an _______________________________________
o
o
o
 shape of a distribution depends on _______________________________
= if population appears Normal, than so is the sampling distribution of x :
individual observations:
sample mean x of SRS size n:
Examples
p 280 11.9: The scores of 12th grade students on the National Assessment of
Educational Progress year 2000 mathematics test have a distribution that is
approximately Normal with mean µ = 300 and standard deviation σ = 35. If you
repeatedly choose an SRS of four 12th graders, what would be the mean and
standard deviation of all the x -values?
p280 11.8: Juan makes a measurement in a chemistry laboratory and records
the result in his lab report. The standard deviation of the students’ lab
measurements is σ = 10 milligrams. Juan repeats the measurement 3 times
and records the mean of his 3 measurements. What is the standard deviation
of Juan’s mean result? (That is, if Juan kept on making 3 measurements and
averaging them, what would be the standard deviation of all his x ’s?)
p296 11.31: The number of lightning strikes on a square kilometer of open
ground in a year has mean 6 and standard deviation 2.4. (The values are
typical of much of the United States.) The National Lightning Detection Network
uses sensors to watch for lightning in a sample of 10 square kilometers. What
are the mean and standard deviation of x , the mean number of strikes per
square kilometer?
Chapter 11 Lesson 3 Homework
For each situation below, find the mean and standard deviation of the mean of the
indicated sampling distribution.
1. The heights of fully grown sugar maple trees are normally distributed, with a mean of 87.5
feet and a standard deviation of 6.25 feet. Random samples of size 12 are drawn from the
population and the mean of each sample is determined.
Repeat the problem for samples of size 24.
Repeat the problem for sample of size 36.
What happens to the mean and standard deviation of the distribution of sample means
as the size of the sample increases?
2. The number of eggs a female house fly lays during her lifetime is normally distributed, with
a mean of 800 eggs and a standard deviation of 100 eggs. Random samples of size 15 are
drawn from this population and the mean of each sample is determined.
3. The per capita consumption of red meat by people in the U.S. in a recent year was
normally distributed, with a mean of 115.6 pounds and a standard deviation of 38.5 pounds.
Random samples of size 20 are drawn from this population and the mean of each sample is
determined.
4. The per capita consumption of soft drinks by people in the U.S. in a recent year was
normally distributed, with a mean of 53.0 gallons and a standard deviation of 17.1 gallons.
Random samples of size 25 are drawn from the population and the mean of each sample is
determined.