1. Distributions of the Sample Mean
Chapter 8 Section 1
Distributions of the
Sample Mean

2. Chapter 8 – Section 1 Learning objectives
Understand the concept of a sampling distribution
Describe the distribution of the sample mean for samples obtained from normal populations
Describe the distribution of the sample mean for samples obtained from a population that is not normal 1 2 3

3. Chapter 8 – Section 1 Learning objectives
Understand the concept of a sampling distribution
Describe the distribution of the sample mean for samples obtained from normal populations
Describe the distribution of the sample mean for samples obtained from a population that is not normal 1 2 3

4. Chapter 8 – Section 1
Often the population is too large to perform a census … so we take a sample
How do the results of the sample apply to the population?
What’s the relationship between the sample mean and the population mean?
What’s the relationship between the sample standard deviation and the population standard deviation?
This is statistical inference
Often the population is too large to perform a census … so we take a sample
Often the population is too large to perform a census … so we take a sample
How do the results of the sample apply to the population?
What’s the relationship between the sample mean and the population mean
What’s the relationship between the sample standard deviation and the population standard deviation?

5. Chapter 8 – Section 1
We want to use the sample mean x to estimate the population mean μ
If we want to estimate the heights of eight year old girls, we can proceed as follows
Randomly select 100 eight year old girls
Compute the sample mean of the 100 heights
Use that as our estimate
This is using the sample mean to estimate the population mean
We want to use the sample mean x to estimate the population mean μ
We want to use the sample mean x to estimate the population mean μ
If we want to estimate the heights of eight year old girls, we can proceed as follows
Randomly select 100 eight year old girls
Compute the sample mean of the 100 heights
Use that as our estimate

6. Chapter 8 – Section 1
However, if we take a series of different random samples
Sample 1 – we compute sample mean x1
Sample 2 – we compute sample mean x2
Sample 3 – we compute sample mean x3
Etc.
Each time we sample, we may get a different result
The sample mean x is a random variable!
However, if we take a series of different random samples
Sample 1 – we compute sample mean x1
Sample 2 – we compute sample mean x2
Sample 3 – we compute sample mean x3
Etc.

7. Chapter 8 – Section 1 Because the sample mean is a random variable
The sample mean has a mean
The sample mean has a standard deviation
The sample mean has a probability distribution
This is called the sampling distribution of the sample mean

8. Chapter 8 – Section 1
When we use the sample mean to estimate the population mean, we are estimating a parameter (number) with a random variable
The sampling distribution of the sample mean has the parameters of
A sample of size n
A population with mean μ
A population with standard deviation σ
When we use the sample mean to estimate the population mean, we are estimating a parameter (number) with a random variable

9. Chapter 8 – Section 1 Example We have the data
1, 7, 11, 12, 17, 17, 17, 21, 21, 21, 22, 22
and we want to take samples of size n = 3
First, a histogram of the entire data set

10. Chapter 8 – Section 1 A histogram of the entire data set
Definitely skewed left … not bell shaped

11. Chapter 8 – Section 1 Taking some samples of size 3
The first sample, 17, 21, 12, has a mean of 16.7

12. Chapter 8 – Section 1
More sample means from more samples

13. Chapter 8 – Section 1
A histogram of 20 sample means

14. Chapter 8 – Section 1
The original data set was skewed left, but the set of sample means is close to bell shaped

15. Chapter 8 – Section 1 Learning objectives
Understand the concept of a sampling distribution
Describe the distribution of the sample mean for samples obtained from normal populations
Describe the distribution of the sample mean for samples obtained from a population that is not normal 3 1 2

16. Chapter 8 – Section 1
If we know that the population has a normal distribution, then the sampling distribution (i.e. the distribution of x) will also be normal
In fact, the sampling distribution
Will be normally distributed
Will have a mean equal to the mean of the population
Will have a standard deviation less than the standard deviation of the population
If we know that the population has a normal distribution, then the sampling distribution (i.e. the distribution of x) will also be normal

17. Chapter 8 – Section 1 Why does it have a smaller standard deviation?
The population standard deviation
Is a measure of the distance between an individual value and the mean
The sampling error for a sample of size n = 1
The standard deviation of the sample mean
Is a measure of the distance between the sample mean and the mean
It makes sense that the estimate is more accurate if we’ve taken more values (a larger n)
Why does it have a smaller standard deviation?
The population standard deviation
Is a measure of the distance between an individual value and the mean
The sampling error for a sample of size n = 1
The standard deviation of the sample mean
Is a measure of the distance between the sample mean and the mean
Why does it have a smaller standard deviation?
The population standard deviation
Is a measure of the distance between an individual value and the mean
The sampling error for a sample of size n = 1
Why does it have a smaller standard deviation?

18. Chapter 8 – Section 1 The Law of Large Numbers says that
As we take more observations to the sample (i.e. as n gets larger), the difference between the sample mean x and the population mean μ approaches 0
That is to say, we get to the right answer with larger and larger sample sizes

19. Chapter 8 – Section 1
The standard error, , is the standard deviation of the sample mean
The formula for is
This is an extremely important formula
The standard error, , is the standard deviation of the sample mean

20. Chapter 8 – Section 1 What does this mean?
If we have a normally distributed random variable X, then the distribution of the sample mean x is completely determined
We know that it’s also normally distributed
We know its mean
We know its standard deviation
So … we can do all of our calculations!

21. Chapter 8 – Section 1
If a simple random sample of size n is drawn from a large population, then the sampling distribution has
Mean and
Standard deviation
In addition, if the population is normally distributed, then
The sampling distribution is normally distributed
If a simple random sample of size n is drawn from a large population, then the sampling distribution has
Mean and
Standard deviation

22. Chapter 8 – Section 1 Example
If the random variable X has a normal distribution with a mean of 20 and a standard deviation of 12
If we choose samples of size n = 4, then the sample mean will have a normal distribution with a mean of 20 and a standard deviation of 6
If we choose samples of size n = 9, then the sample mean will have a normal distribution with a mean of 20 and a standard deviation of 4

23. Chapter 8 – Section 1 Learning objectives
Understand the concept of a sampling distribution
Describe the distribution of the sample mean for samples obtained from normal populations
Describe the distribution of the sample mean for samples obtained from a population that is not normal 1 2 3

24. Chapter 8 – Section 1
This is great if our random variable X has a normal distribution
However … what if X does not have a normal distribution
What can we do?
Wouldn’t it be very nice if the sampling distribution for X also was normal?
This is almost true …

25. Chapter 8 – Section 1 The Central Limit Theorem states
Regardless of the shape of the distribution, the sampling distribution becomes approximately normal as the sample size n increases
Thus
If the random variable X is normally distributed, then the sampling distribution is normally distributed also
For all other random variables X, the sampling distributions are approximately normally distributed
The Central Limit Theorem states
Regardless of the shape of the distribution, the sampling distribution becomes approximately normal as the sample size n increases

26. Chapter 8 – Section 1
This approximation, of the sampling distribution being normal, is good for large sample sizes … large values of n
How large does n have to be?
A rule of thumb – if n is 30 or higher, this approximation is probably pretty good

27. Chapter 8 – Section 1 Example
We’ve been told that the average weight of giraffes is 2400 pounds with a standard deviation of 300 pounds
We’ve measured 50 giraffes and found that the sample mean was 2600 pounds
Is our data consistent with what we’ve been told?

28. Chapter 8 – Section 1
The sample mean is approximately normal with mean 2400 (the same as the population) and a standard deviation of 300 / √ 50 = 42.4
Using our calculations for the general normal distribution, 2600 is 200 pounds over 2400, and 200 pounds is 200 / 42.4 = 4.7
From our normal calculator, there is about a 1 chance in 1 million of this occurring
Something is definitely strange … we’ll see what to do later in inferential statistics
The sample mean is approximately normal with mean 2400 (the same as the population) and a standard deviation of 300 / √ 50 = 42.4
Using our calculations for the general normal distribution, 2600 is 200 pounds over 2400, and 200 pounds is 200 / 42.4 = 4.7
From our normal calculator, there is about a 1 chance in 1 million of this occurring
The sample mean is approximately normal with mean 2400 (the same as the population) and a standard deviation of 300 / √ 50 = 42.4
Using our calculations for the general normal distribution, 2600 is 200 pounds over 2400, and 200 pounds is 200 / 42.4 = 4.7
The sample mean is approximately normal with mean 2400 (the same as the population) and a standard deviation of 300 / √ 50 = 42.4

29. Summary: Chapter 8 – Section 1
The sample mean is a random variable with a distribution called the sampling distribution
If the sample size n is sufficiently large (30 or more is a good rule of thumb), then this distribution is approximately normal
The mean of the sampling distribution is equal to the mean of the population
The standard deviation of the sampling distribution is equal to

30. Chapter 8 – Example 1
The combined (verbal + quantitative reasoning) score on the GRE is normally distributed with mean 1066 and standard deviation 191. (Source: Suppose n = 15 randomly selected students take the GRE on the same day.
a. What is the probability that a randomly selected student scores above 1100 on the GRE?
b. Describe the sampling distribution of the sample mean.
c. What is the probability that a random sample of 15 students has a mean GRE score that is less than 1100?
d. What is the probability that a random sample of 15 students has a mean GRE score that is 1100 or above?

31. Chapter 8 – Example 1
The combined (verbal + quantitative reasoning) score on the GRE is normally distributed with mean 1066 and standard deviation 191. (Source: Suppose n = 15 randomly selected students take the GRE on the same day.
a. What is the probability that a randomly selected student scores above 1100 on the GRE? (0.4286)
b. Describe the sampling distribution of the sample mean. (It is normal with mean 1066 and standard deviation 49.3)
c. What is the probability that a random sample of 15 students has a mean GRE score that is less than 1100? (0.7549)
d. What is the probability that a random sample of 15 students has a mean GRE score that is 1100 or above? (0.2451)

32. Chapter 8 – Example 2
In the United States, the year each coin was minted is printed on the coin. To find the age of a coin, simply subtract the current year from the year printed on the coin. The ages of circulating pennies are right skewed. Assume the ages of circulating pennies have a mean of 12.2 years and a standard deviation of 9.9 years.
a. Based on the information given, can we determine the probability that a randomly selected penny is over 10 years old?
b. What is the probability that a random sample of 40 circulating pennies has a mean less than 10 years?
c. What is the probability that a random sample of 40 circulating pennies has a mean greater than 10 years?
d. What is the probability that a random sample of 40 circulating pennies has a mean greater than 15 years? Would this be unusual?

33. Chapter 8 – Example 2
In the United States, the year each coin was minted is printed on the coin. To find the age of a coin, simply subtract the current year from the year printed on the coin. The ages of circulating pennies are right skewed. Assume the ages of circulating pennies have a mean of 12.2 years and a standard deviation of 9.9 years.
a. Based on the information given, can we determine the probability that a randomly selected penny is over 10 years old? (No, because the population of the ages of circulating pennies is not normally distributed.)
b. What is the probability that a random sample of 40 circulating pennies has a mean less than 10 years? (0.0793)
c. What is the probability that a random sample of 40 circulating pennies has a mean greater than 10 years? (0.9207)
d. What is the probability that a random sample of 40 circulating pennies has a mean greater than 15 years? Would this be unusual? (0.0367; yes)