1. Sampling Distribution Models
Population – all items
of interest.
Population
Parameter
Inference
We are ready to start putting together the pieces to come up with how to use sample statistics to make inferences, educated guesses, about population parameters. Remember our picture of the population and sample. We will not complete the process by introducing the idea of using a sample statistics as an estimate of the population parameter. But before that we need to know a little more about how sample statistics based on random samples from a population behave.
Sample – a
few items from
the population.
Sample
Statistic
Random
selection

2. Proportions
So far we have used the sample proportion, , to make inferences about the population proportion p.
To do this we needed the distribution of .

3. Distribution of Shape: Approximately Normal if conditions are met.
Center: The mean is p.
Spread: The standard deviation is

4. Categorical Variable
When the response variable of interest is categorical, the parameter is the proportion of the population that falls in a particular category, p.

5. Quantitative Variable
When the response variable of interest is quantitative, the parameter is the mean of the population, .

6. Means We will use the sample mean,
, to make inferences about the population mean, .
To do this we need the distribution of .

7. Simulation

8. Simulation Simple random sample of size n=5. Repeat many times.
Record the sample mean, , to simulate the distribution of .

9. Simulation Different samples will produce different sample means.
There is variation in the sample means.
Can we model this variation?

11. Population Shape: Basically normal Center: Mean,
Spread: Standard Deviation,

12. Distribution of n = 5 Shape: Normal Center: Mean,
Spread: Standard Deviation,

14. Population Shape: Not normal, skewed right Center: Mean,
Spread: Standard Deviation,

15. Distribution of n = 5 Shape: Approximately normal Center: Mean,
Spread: Standard Deviation,

17. Population Shape: Not normal, skewed right Center: Mean,
Spread: Standard Deviation,

18. Distribution of n = 25 Shape: Approximately normal Center: Mean,
Spread: Standard Deviation,

19. Central Limit Theorem
When selecting random samples from a population with a distribution that is not normal, the sampling distribution of will be approximately normally distributed.
The larger the sample the better the approximation.

20. Conditions Random sampling condition 10% condition
Samples must be selected at random from the population.
10% condition
When sampling without replacement, the sample size should be less than 10% of the population size.

21. Summary
Distribution of
Shape: Approximately normal
Center:
Spread: