1. Sampling Distribution Models
Chapter 18
Sampling Distribution Models

2. Topics The sampling distribution of a proportion/mean.
The mean and standard deviation of a sampling distribution of a proportion/mean.
Normality revisited
Necessary conditions
Standard Error

3. Presidential Election 1996
In the 1996 presidential election, Bill Clinton received 49% of the popular vote, Bob Dole received 41% of the popular vote, and Ross Perot received 8% of the popular vote.
If we were to sample the public regarding how they voted (or were planning to vote), what would we expect to obtain?
How accurate would our sample proportion be?
How likely would we be to get a sample predicting Bob Dole would win the election?
These are the types of questions we will answer in related to the sampling distribution of the mean.

4. Election example ctd.
Run a simulation in which we randomly sample 200 voters and ask each of them their presidential preference. We are interested in finding the percentage of the voters that would choose Bob Dole. Perform 1000 runs to simulate choosing 1000 different samples of size 200.
What is the shape of our distribution?
What does each data value (or dot) represent?
What is the mean of the distribution?
What is the standard deviation of the distribution?

5. Election example ctd.
What is the probability of choosing a sample predicting that Bob Dole would get more than 50% of the vote?
What is the probability of choosing a sample predicting that Dole would get between 40% and 42% of the vote?
What would we expect to happen if we increased our sample size to 500? To 10,000?

6. The sampling distribution of a proportion
The Sampling Distribution of a Proportion is the set of all possible sample proportions of size n.
Notation: is the notation for a sample proportion. It is also used to represent the sampling distribution of a proportion. The difference can be made clear by context.
The mean of the sampling distribution:
The standard deviation of the sampling dist:

7. Presidential Example ctd.
Returning to the Dole example,
What is the true mean of the sampling distribution of the proportion for samples of size 200?
What is the true standard deviation of the sampling distribution of the proportion for samples of size 200?
What is the actual probability of obtaining a random sample of size 200 in which Dole obtains more than 50% of the vote?

8. Assumptions and Conditions for Normality (The Central Limit Theorem)
What does it take for the sampling distribution to normal?
1) The sampled values must be chosen independently of each other (with replacement).
2) The sample size must be large enough.
10% condition: If the samples are not obtained with replacement, the sample size must not exceed more than 10% of the population size.
Success/Failure Conditions: The sample size has to be large enough so that both np and nq = n(1-p) are at least 10.
Example: How large does the sample size have to be in the presidential election when analyzing Bob Dole’s proportion of the vote?

9. Quantitative Data and the Sampling Distribution of the Mean
If we are analyzing quantitative data, and not necessarily interested in a proportion having a specified attribute, we may be interested in the average value we would expect to obtain.
The sampling distribution of the mean is the set of all sample means of size n from a population.

10. Sampling Distribution of the Mean example
Suppose a kennel has 5 dogs with the following weights: 30, 36, 48, 60, 72. (pounds)
Construct the sample distribution of the mean for samples of size 2.
Construct the sample distribution of the mean for samples of size 3.

11. Example ctd. What is the mean of the sample distribution of the mean?
What happens to the spread of the sampling distribution of the mean as the sample size grows larger?

12. Mean and Standard Deviation of The sampling distribution
For a sampling distribution of the mean,
What happens to the standard deviation of the sampling distribution as the sample size increases?

13. Example
According to a recent study, the average salary of advanced degree (post Master’s degrees) holders is $42,000 with a standard deviation of $6,000.
For samples of size 36, determine the mean and standard deviation of the sampling distribution of the mean.
Interpret this result.
What about samples of size 100?

14. Notation note
The variable is frequently used to denote the sample distribution of the mean.
This is the same notation as individual sample means, so you determine which is being talked about by the context.

15. When can we assume that x is normal? (Central Limit Theorem)
1) When the original variable under consideration is normal.
2) When the sample size is 30 or greater.

16. Conditions
Random sampling condition—The data values must be randomly sampled.
Independence Assumption—the sampled values should be mutually independent. If the sampling is done without replacement (this is usually the case), then the sample size should not exceed 10% of the population size (Usually not a problem).

17. Example
A variable under consideration has a mean of 50 and a standard deviation of 20.
A) Identify the sampling distribution of the mean for samples of size 36.
B) What can you say about the sample distribution of the mean for samples of size 25?

18. Empirical Rule
Recall that 68.26% of all data values in a normal distribution lie within 1 standard deviation of the mean.
95.44% of all data values in a normal distribution lie with 2 standard deviations of the mean.
99.74% of all data values in a normal distribution lie within 3 standard deviations of the mean.

19. What are data values in the sampling distribution of the mean?
Data values are sample means of all the possible samples of size n.

20. 68.26% of all possible samples of size n have means that lie within of the actual population mean.

21. Equivalent Interpretations
68.26% of all samples of size n have the property that the population mean is contained in the interval from

22. 95.44% of all samples of size n have the property that the population mean is contained in the interval from

23. 99.74% of all samples of size n have the property that the population mean is contained in the interval

24. Example
The average commuting time in a city 40 minutes with a standard deviation of 10 minutes. Commuting time may or may not be a normal variable. Suppose a sample of 100 commuters is studied. Determine whether the following statements are true or false.
A) There is a roughly 68.26% chance that the mean of the sample will be between 30 and 50.
B) % of all possible observations of x lie between 30 and 50.
C) There is roughly a 68.26% chance that the mean of the sample will be between 39 and 41.
What if commuting time is a normal variable?

25. Standard Error What if we don’t know p or σ?
We would expect to use the sample values instead.
The standard error for the sampling distribution of the proportion:
The standard error for the sampling distribution of the mean: