Chapter 9: Sampling DistributionsParameter: a number that describes the population. (In practical statistics, this value is often unknown)

Statistic: a number that can be computed from the sample data without making use of any unknown parameter.

Statistic is commonly either a mean () or a proportion ( )

Parameters are symbolized by and .

Parameter or statistic?

The mean income of the sample of households contacted by the Current Population Survey was $60,528.

Example: The Gallup Poll asked a random sample of 515 US adults whether they believe in ghosts. Of the respondents, 160 said “yes”. What is the proportion of US adults that believe in ghosts?

31.0515160ˆ p

The table below shows the scores of a recent statistics test for 10 students.

The parameter of interest is the mean score in this population, which is 69.4. The sample is an SRS drawn from the population.

1) Use a random digit table (numbering students from 0 – 9) to select a SRS of 4 from this population. Write the 4 scores and find the mean. This statistic is an estimate for the mean of the population.

2) Repeat this a total of 10 times and create a histogram of the mean scores for each sample.

3) Find the mean of your mean scores.

Student #

1 2 3 4 5 6 7 8 9 10

Score 82 62 80 58 72 73 65 66 74 62

So how can we use the sample statistic to estimate the parameter for the population?

- Take a large number of samples- Calculate the desired statistic for each sample- Create a histogram of the statistics- Examine the distribution (shape, center, spread, etc.)

This process will approximate a distribution of statistic values in all possible samples of the same size from the same population (called a sampling distribution)

How trustworthy is this statistic as an estimate of the parameter? (see pages 573 – 576)

Variability of a statistic is also a concern. This is described by the spread of the sampling distribution and depends upon the sampling design and sample size.

Larger sample size smaller spread (less variability)

If population ≥ 10(sample size), then

spread of sampling distribution ≈ spread of population

- A statistic is said to be unbiased if the mean (center) of the sampling distribution is equal to the parameter value. - The statistic is referred to as an unbiased estimator.- We can tell that a statistic is unbiased if the distribution is approximately normal.

Variability

Bias

HIGH LOW

HIGH-Center of distribution ≠ parameter (non-symmetric)-Large spread (lots or spread out bars)

-Center of distribution ≠ parameter (non-symmetric)-Small spread (few or compacted bars)

LOW-Center of distribution ≈ parameter (symmetric)-Large spread (lots or spread out bars)

-Center of distribution ≈ parameter (symmetric)-Small spread (few or compacted bars)

↑ Desired outcome