Transcript 9.3.1 - GEOCITIES.ws

The Distribution of Sample
Means
Section 9.3.1
Starter 9.3.1
•
Write a sentence or two that answers
each question:
1. What is a sample proportion?
2. What is a sampling distribution?
3. What is the significance of the mean of a
sampling distribution of sample proportions?
Fathom Demo
• Fathom Demo 20 (The Distribution of
Sample Proportions) shows the
connection between:
– Sample size and the spread of a sampling
distribution
– True population proportion and the center of a
sampling distribution
Today’s Objectives
• Find the mean and standard deviation of the
sampling distribution of x (sample means) for a
given sample size n
• Explain why averages are less variable than
individual observations
California Standard 16.0
Students know basic facts concerning the relation
between the mean and the standard deviation of a
sampling distribution and the mean and the standard
deviation of the population distribution.
Distribution of Sample Means
• Suppose we draw an SRS of size n from a
normally distributed population with mean µ and
standard deviation σ.
• Then the sampling distribution is:
– Shape: approximately normal
– Center:
– Spread:
X  
x 

n
• Note: The population must be at least 10 times sample size
to use this formula
Example 9.7
• Suppose it is known that the heights of young
women are N(64.5 in, 2.5 in).
• Simulate measuring the heights of an SRS of
100 young women in your calculator.
– randNorm(64.5, 2.5, 100)  L1
• Report your results to me. Put class results in L2
– Run 1-Var Stats to find mean and s.d. of
• What is the theoretical distribution of
x
x
?
– The mean of the sampling distribution should be 64.5
– The standard deviation should be .25
Continue the problem
•
•
•
•
If the sample mean you measured turned out to
be 63.9 in, what is the probability of finding a
group of 100 young women that short or shorter?
normalcdf(0, 63.9, 64.5, .25) = .008
There was a 0.8% chance of getting these results
if the population distribution was correct, so I find
the results surprising.
Possible reasons:
1. Perhaps sampling methodology was flawed.
2. Perhaps the original population distribution was wrong
3. Perhaps the 0.8% chance really happened!
How do averages compare to
individual observations?
• Individual observations can vary widely.
Consider the heights of Yao Ming and Verne
Troyer (Mini-me). But when we look at several
individuals and average the results, the high
values tend to balance the low values. Thus
averages don’t have such extremes: they are
less variable than individual observations.
• Now look again at the standard deviation
formula:

x 
n
Today’s Objectives
• Find the mean and standard deviation of the
sampling distribution of x (sample means) for a
given sample size n
• Explain why averages are less variable than
individual observations
California Standard 16.0
Students know basic facts concerning the relation
between the mean and the standard deviation of a
sampling distribution and the mean and the standard
deviation of the population distribution.
Homework
• Read pages 481 - 484
• Do problems 26 - 29