Chapter 9 Sampling Distributions 9 1 Sampling Distributions

Chapter 9 Sampling Distributions 9. 1 Sampling Distributions

Parameter vs. Statistic • A parameter is a number that describes the population. In statistical practice, the value of a parameter is not known. • A statistic is a number that can be computed from the sample data without making use of any unknown parameters. In practice, we often use a statistic to estimate an unknown parameter.

Sampling Variability • The value of a statistic varies in repeated random sampling. • is a statistic – a sample proportion • is a parameter

“What would happen if we took many samples? ” • 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 the sample means or sample proportions. • Examine the distribution displayed in the histogram for shape, center, and spread as well as for outliers or other deviations.

Using Simulation • In practice, it is too expensive to take many samples from a large population. • We can imitate many samples by using simulation. • Example 9. 3 page 566

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. • It is the ideal pattern that would emerge if we looked at all possible samples of a certain size from our population.

Exercises • • Assignment p. 568 problems 9. 1 and 9. 2 Page 570 problem 9. 5 Yes, I want you to repeat it 10 times! Combine all data from the class

Describing Sampling Distributions • • Overall shape Outliers Center Spread

The Bias of a Statistic • Bias concerns the center of the sampling distribution. • A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. • The statistic is called an unbiased estimator.

The Variability of a Statistic • The variability of a statistic is described by the spread of its sampling distribution. This spread is determined by the sampling design and the size of the sample. Larger samples give smaller spread. • As long as the population is much larger than the sample (say, at least 10 times as large), the spread of the sampling distribution is approximately the same for any population size.