1. Section 7.1 Sampling Distributions

2. Vocabulary Lesson Parameter Statistic
A number that describes the population.
This number is fixed. In reality, we do not know its value because we can’t examine the WHOLE population.
Statistic
A number that describes a sample.
We know this value because we compute it from our sample, but it can change from sample to sample.
We use a statistic to estimate the value of a parameter.

3. Population – Parameter
Helpful hints
Sample – Statistic
Population – Parameter

4. Notation
Parameter
Statistic
Mean
Standard Deviation s
Proportion p

5. Example What is the mean income of households in the United States?
The government’s Current Population Survey contacted a sample of 50,000 households in The mean income of this sample was $57,045.
What is the parameter of interest?
Is $57,045 a parameter or a statistic?
If we took another sample of 50,000 households, would μ change? What about ?

6. Example 2
A carload of ball bearings has mean diameter cm. This is within the specifications for acceptance of the lot by the purchaser. By chance, an inspector chooses 100 bearings from the lot that have mean diameter cm. Because this is outside the specified limits, the lot is mistakenly rejected.
State whether the highlighted numbers are parameters or statistics. Use correct notation.
What is the population?

7. Sampling Variability
If we took repeated samples of the households in the United States, each one would likely produce a different sample mean.
That’s called sampling variability:
The value of a statistic varies in repeated random sampling.

8. What would happen if we took many samples?
If you are conducting a simulation, you should:
Take a large number of samples from the same population.
Calculate the sample mean or sample proportion for each sample.
Make a histogram of the values of x-bar or p-hat.
Examine the distribution displayed in the histogram for shape, center, and spread.

9. Revisiting IQ
Let’s simulate taking a random sample size 25 from a normal population (100,15). We’ll calculate the sample mean for each sample and record the result. We’ll repeat this 20 times. We could, but we aren’t!
We would have found the sampling distribution.
The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

10. Recall… What is the difference between a parameter and a statistic?
Which symbol do I use for the following?
Population proportion?
Sample mean?
Population mean?
Sample standard deviation?
Sample proportion?
Population standard deviation?

11. Sampling Distributions
A sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the population.

12. Describing Sampling Distributions
We can use the tools of data analysis to describe any distribution, including a sampling distribution.
SOCS

13. Example
Let the population be all the numbers on a die. Roll two dice (an SRS of two) and find the average value for each set of rolls. Complete this 20 times and make a histogram. Describe the histogram.
Can we construct the ACTUAL probability distribution? List all possible outcomes and their x-bars. If we find the mean of the x-bars, then this is the actual mean.

14. So…

15. Bias
So far we’ve talked about the bias of a sampling method. However, it is often useful to talk about the bias of statistic. When talking about a statistic, bias concerns the center of the sampling distribution.

16. The bias of a statistic
When a sampling distribution centers around the true value of the parameter, we say it is unbiased.
In other words, a statistic is unbiased if the mean of its sampling distribution equals the true value of the parameter being estimated.
There is no SYSTEMATIC tendency to under- or overestimate the value of the parameter.

17. The variability of a statistic
The variability of a statistic is described by the spread of its sampling distribution.
LARGER SAMPLES GIVE SMALLER SPREAD.
Notice that this is saying that larger samples are good. However, it says NOTHING about the size of the population.
Q: Does the size of the population matter? A: No. We usually require the population be ten times larger than the sample. So if n = 20, population should be at least 200.

18. A Note about Population Size
Example: Suppose the Mars Company wants to check that their M&Ms are coming out properly (i.e. Not broken, not undersized, etc.).
It doesn’t matter if you select a random scoop from a truckload or a large bin. (Meaning: population size doesn’t matter)
As long as the scoop is selecting a random, well-mixed sample, we’ll get a good picture of the quality of M&Ms.

19. Bias vs. Variability What does the bull’s-eye represent?
What do the darts represent?

20. Homework
Chapter 7
#1-4, 5-8, 18, 19