Download Section 8.1 –Distribution of the Sample Mean

Section 8.1 –Distribution of the Sample Mean
Objectives
1. Describe the distribution of the sample mean: normal population
2. Describe the distribution of the sample mean: non-normal population
Statistics such as x are random variables since their value varies from sample to sample. As such,
they have probability distributions associated with them. In this section we focus on the shape,
center and spread of x . (What is a random variable?)
Suppose we have three pool balls, each with a number on it. Two of the balls are selected randomly
(with replacement) and the average of their numbers is computed.
Create a table listing all possible outcomes.
Trial Ball 1
1
2
3
4
5
6
7
8
9
Ball 2
Mean
Now form a frequency and relative frequency table for the mean.
Mean Frequency Relative Frequency
Now draw a relative frequency distribution of the results above.
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This distribution is also a probability distribution since the Y-axis is the probability of obtaining a given
mean from a sample of two balls in addition to being the relative frequency. (What are the two
criteria a probability distribution must satisfy?)
The distribution shown above is called the sampling distribution of the mean, i.e., it is the sampling
distribution of the mean for a sample size of 2 (n = 2). For this simple example, the distribution of
pool balls and the sampling distribution are both discrete distributions. The pool balls have only the
values 1, 2, and 3, and a sample mean can have one of only five values we calculated above.
We could have obtained the relative frequency distribution above another way. How?
The idea behind obtaining the sampling distribution of the mean is as follows:
Step 1: Obtain a simple random sample of size n.
Step 2: Compute the sample mean.
Step 3: Assuming we are sampling from a finite population, repeat Steps 1 and 2 until all
simple random samples of size n have been obtained.
But what if we wanted to repeat the above with 1000 pool balls numbered 1 – 1000? Or a million
pool balls? Or any truly continuous distribution? Creating a table of all possible outcomes is not
practical.
Sampling Distribution of a Statistic – A probability distribution for all possible values of the statistic
computed from a sample of size n.
Sampling Distribution of the Sample Mean x – A probability distribution of all possible values of
x computed from a sample of size n from a population with mean µ and standard deviation  .
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It is important to keep in mind that every statistic, not just the mean, has a sampling distribution. For
this chapter we will focus only on the sampling distribution of the mean.
For the applications in this section,
• When working with an individual value from a normally distributed population, use z 
x  

i.e., normalcdf (lower bound, upper bound, µ, σ)
• When working with the mean of a sample (group), use z =
i.e., normalcdf (lower bound, upper bound, µ,

n
x - 




n

)
Objective 1 – Describe the Distribution of the Sample Mean: Normal Population
In other words, take samples from a population that is normally distributed, calculate the mean of
each sample, and examine the distribution of the sample means.
If the original population is normally distributed, then the distribution of the sample means will be
normally distributed, regardless of the sample size, n.
Open StatCrunch, click Applets -> Sampling Distributions
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In the window that opens, select “Bell shaped” and then click “Compute” to get the window below.
In the Sample size box, where the above has a 30, enter a 5 then click “1 time”. Continue to click “1
time” for say 20 times. What happens in the “Sample means” distribution at the bottom of the
window?
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Example
The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46
grams and standard deviation 0.02 grams. What is the probability that in a simple random sample of
10 pennies minted after 1982, we obtain a sample mean of at least 2.465 grams?
Example
Scores for men on the verbal section of the SAT test are normally distributed with a mean of 509 and
a standard deviation of 112.
a) If one man is chosen, find the probability that he scores less than 590.
b) If 16 men are chosen, find the probability that their mean score is less than 590.
Objective 2 – Describe the Distribution of the Sample Mean: Non-Normal Population
If the original population is not normally distributed, then the distribution of the sample means will
become approximately normally distributed as n increases.
Again in StatCrunch, click Applets -> Sampling Distribution, but in the resulting window select Right
skewed and click Compute.
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You should see this window…
Notice how the parent population is skewed right. Experiment with the Sample size, where the 2 is,
and run the experiment a number of times and observe what happens with the “Sample means.” See
if you can validate the Central Limit Theorem.
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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.
a) If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the
sample mean.
b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil
change time is less than 11 minutes?
Key Points from Section 8.1

The mean of the sampling distribution is equal to the mean of the parent population and the
standard deviation of the sampling distribution of the sample mean is

n
regardless of the
sample size.

The Central Limit Theorem: the shape of the distribution of the sample mean becomes
approximately normal as the sample size n increases, regardless of the shape of the
population.
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