1. Modular 13
Ch 8.1 to 8.2

2. Ch 8.1 Distribution of the Sample Mean
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed
Ch Distribution of the Sample Proportion
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed

3. Ch 8.1 Distribution of the Sample Mean
Objective A : Shape, Center, and Spread of the Distribution of
A1. Sampling Distributions of Mean
Assume equal chances for each number to be selected.
This population distribution of is normally distributed.

4. Sampling Probability distribution of mean .
Let’s say we select two elements ( ) from with replacement. (Independent case)
List out all possible combinations (sample space) and for each combination.

5. Probability distribution of is summarized in the table shown below.
Probability histogram for .

6. Let’s compare the distribution shape of and .
The population distribution of is uniformly distributed .
The sampling distribution of is normally distributed.
Is this by chance that is normally distributed?

7. A2. Central Limit Theorem
A. If the population distribution of is normally distributed, the sampling distribution of is normally distributed regardless of the sample size .
If the population distribution is not normally distributed, the sampling distribution of is guaranteed to be normally distributed if
B. Mean/standard deviation of a sampling distribution of vs mean/standard deviation of a population distribution of .
The mean and standard deviation of population distribution are and respectively.
The mean of the sampling distribution of is
The standard deviation of the sampling distribution of is

8. Example 1 : Determine and from the given parameters of the population and the sample size.

9. Example 2 : A simple random sample of is obtained from a population with and .
(a) If the population distribution is skewed to the right, what condition must be applied in order to guarantee the sampling distribution of is normally distributed?
Since the population distribution is not normally distributed, the selected sample size must be greater than or equal to 30 (i.e ).
(b) If the sample size is , what must be true regarding the distribution of the population in order to guarantee the sampling distribution of to be normally distributed?
For small sample size, the population distribution must be normally distributed in order to guarantee the sampling distribution of to be normally distributed.

10. Ch 8.1 Distribution of the Sample Mean
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed
Ch Distribution of the Sample Proportion
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed

11. Ch 8.1 Distribution of the Sample Mean
Objective B : Finding Probability of that is Normally Distributed
Standardize to
Recall : Standardize to :
Now : Standardize to :

12. Example 1 : A simple random sample of size is obtained from a population mean and population standard deviation
(a) Describe the sampling distribution .
Since , is normally distributed.
(b) What is ?

13. From Table V

14. Example 2: The upper leg of 20 to 29 year old males is normally distributed with a mean length of 43.7cm and a standard deviation of 4.2cm.
(a) What is the probability that a random sample of 12 males who are to 29 years old results in a mean upper leg length that is between 42cm and 48cm?
Population is normally distributed.
Since the population distribution is normally distributed, is normally distributed for any sample size.

15. From Table V

16. (b) A random sample of 15 males who are 20 to 29 years old results in a mean upper leg length greater than 46 cm. Do you find the result unusual? Why?
In order to know if it is unusual or not, we need to find the probability.
From Table V
Since is smaller than 0.05, this result is unusual.

17. Ch 8.1 Distribution of the Sample Mean
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed
Ch Distribution of the Sample Proportion
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed

18. Ch 8.2 Distribution of the Sample Proportion
Objective A : Shape, Center, and Spread of the Distribution of
Distribution of the Sample Proportions
A. Sampling distribution of sample proportion , where
The shape of the sampling distribution of is approximately normally provided by, or
where
B. Finding the mean and standard deviation of

19. Ch 8.1 Distribution of the Sample Mean
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed
Ch Distribution of the Sample Proportion
Objective A : Shape, Center, and Spread of the Distribution of
Objective B : Finding Probability of that is Normally Distributed

20. Ch 8.2 Distribution of the Sample Proportion
Objective B : Finding Probability of that is Normally Distributed
Standardize to
where and
provided is approximately normally distributed.

21. Example 1: A nationwide study indicated that 80% of college students who use a cell phone, send and receive text messages on their phone. A simple random sample of college students using a cell phone is obtained.
(a) Describe sampling distribution of
Check to see if
Since , is normally distributed.

22. (b) What is the probability that 154 or fewer college students in the sample send and receive text messages on the cell phone? Is this unusual?
Sample proportion :
Looking for probability :
Standardize to
From Table V
Since is bigger than 0.05, this result is usual.