Section 7.1 Sampling Distributions

Vocabulary Lesson Parameter 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.

Helpful hints Sample – Statistic Population – Parameter

Notation ParameterStatistic Mean Standard Deviation s Proportion p

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 ?

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?

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.

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.

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

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.

Heights Put your height in inches on the front board. We will randomly choose 5 students at a time to look at the average of the heights in this class. Look at averages of 20 samples from the class. Is this a statistic or parameter? Create a histogram of the averages you found. What is the average height of the whole class? Is this a statistic or parameter? How does it compare to the center of your histogram? Is this an actual sampling distribution?

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.

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. Was our sampling distribution of heights biased or unbiased?

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.

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.

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

Homework p. 429 (6, 8, 18, 21, 22, 23, 24)