# Chapter 7 Sample Variability Statistics I

```Statistics I
MTH160
Chapter 7
Sample Variability
7.1
Sampling Distribution
7.2
The Sampling Distribution of Sample Means
7.3
Application of the Sampling Distribution of Sample Means
MTH 160
Brigitte Martineau
Statistics I
Chapter 7
7.1
Sampling Distribution
Do you think the sample mean varies from sample to sample?
What is a Sampling Distribution of a Sample Statistics?
The distribution of values for a sample statistics obtained from ____________ samples,
all of the same __________ and all drawn from the same _____________________.
Example:
Consider the population of even digit on a die: {2, 4, 6}

Find all samples of size 2 with their respective mean.
Samples
Mean
What is the probability of selecting any of these samples?
We say that each of these samples is ___________ ________________
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MTH 160
Brigitte Martineau

Statistics I
Chapter 7
Construct the sampling distribution for the sample mean of samples of size 2.
Sample Means

Probabilities
Construct a histogram of the probability distribution.
We just created a sampling distribution of sample means also called the sampling distribution
of x .
We could also create a sampling distribution of sample range, or sample minimum,…
Let’s look at the sampling distribution of sample range
Sample Ranges
Probabilities
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MTH 160
Brigitte Martineau
7.2
Statistics I
Chapter 7
The Sampling Distribution of Sample Means
The Sampling Distribution of Sample Means (SDSM)
If all possible random samples, each of size n, are taken from any population with mean μ and
a standard deviation σ,

The mean of the sampling distribution of x is equal to

The standard deviation of the sampling distribution of x is equal to

If the sampled population has a normal distribution then the sampling distribution of x
will
The Central Limit Theorem (CLT)
If the sampled population is not normal or unknown, the sampling distribution of sample means
will more closely resemble…
Look at p. 373 and 374 for illustrations of the CLT
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MTH 160
Brigitte Martineau
Statistics I
Chapter 7
Examples:

Suppose x has a normal distribution with mean   18 and standard deviation   3 . If
we draw random samples of size 5 from the x distribution and x represents the sample
mean, what can you say about the x distribution?

Suppose x has a mean   75 and a standard deviation   12 but we have no
information as to whether or not the x distribution is normal. If we draw random samples
of size 30 from the x distribution and x represents the sample mean, what can you say

Suppose you did not know that x had a normal distribution. Would you be justified in
saying that the x distribution is approximately normal if the sample size was n  8
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MTH 160
Brigitte Martineau
7.3
Statistics I
Chapter 7
Applications of the Sampling Distribution of Sample Means
When the sampling distribution of sample means is normally distributed (or approximately
normal by CLT) we can answer probability question using the standard normal distribution.
But how can we transform our x distribution into a z distribution?
z
x  x
x

Examples

A random sample of size 36 is to be selected from a population that has a mean of 50
and a standard deviation of 10.
o This sample of 36 has a mean value of x that belongs to a sampling distribution.
Find the shape of this sampling distribution.
o Find the mean of this sampling distribution.
o Find the standard error of this sampling distribution.
o What is the probability that this sample mean will be between 45 and 55?
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MTH 160
Brigitte Martineau
Statistics I
Chapter 7
o What is the probability that this sample mean will have a value greater than 48?

Consider a normal population with   100 and   20 . Suppose a sample of size 16 is
selected at random.
o
P(90  x  110)
o
P( x  115)
o
P( x  115)
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MTH 160
Brigitte Martineau

Statistics I
Chapter 7
The amount of fill (weight of contents) put into a glass jar of spaghetti sauce is normally
distributed with mean   850g and standard deviation   8g
o Describe the distribution of x, the amount of fill per jar.
o Describe the distribution of x , the mean weight for a sample of 24 such jars of
sauce.
o Find the probability that one jar selected at random contains between 848g and
855g.
o Find the probability that a random sample of 24 jars has a mean weight between
848g and 855g
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