Sampling Distribution

Definitions

Central Limit Theorem: A sampling distribution of the mean is approximately normally distributed if the sample size is sufficiently large. This is true no matter what the population distribution is.

Sampling Distribution: a distribution of all of the possible values of a sample statistic for a given sample size selected from a population.

Standard Error: the standard deviation of the sampling distribution.

Unbiased Estimator: Statistic ω is an unbiased estimator of parameter Ω if E(ω) = Ω.

These are the illustrations I have used in class:

The Binomial Distribution Applet from Matt Bognar's (Department of Statistics and Actuarial Science, University of Iowa) page.

• Try to enter several n and p pairs such that either np or np(1-p) is less than 5. Does the distribution look normal?
• Try to enter several n and p pairs such that both np and np(1-p) are no less than 5. Does the distribution look normal?
• NOTE: in this applet, p is the population proportion (i.e., the probability that the event of interest happens during a single random trial, or a single item randomly selected has the characteristic of interest). We (and the textbook) called it π.

Notes

If you draw many samples from a population, they are not going to be all same. And if you calculate the means of various samples, those means will differ. In other words, the sample mean is a random variable. As such, it has a distribution. That is also the case for the sample proportion and many other statistics.

• Population distribution gives probabilities for all possible values in a population
• Parameter distribution is super simple. It's a single value that happens with probability 1 (all other values' probabilities = 0)
• Sampling distribution gives probabilities for the means of all possible samples (of same size) from a population

Try Sampling Distribution Simulator to see how the sampling distribution looks like for various populations.

Read These

Chapter 7. Sampling Distributions in the textbook:

7.1 Sampling Distributions (p. 251)

7.2 Sampling Distribution of the Mean (pp. 251-261)

7.3 Sampling Distribution of the Proportion (pp. 262-264)

You may omit section 7.4 Sampling from Finite Populations (p. 265)

Watch This

Answer These

The amount of baggage a plane passenger checks is random, with a mean of 20 lbs and a standard deviation of 30 pounds. Can you calculate the probability that a passenger checks more than 30 pounds of baggage? If yes, explain how. If no, explain why not.

The company JCrew advertises that 95% of its online orders ship within two working days. You select a random sample of 200 of the 10,000 orders received over the past month to audit. The audit reveals that 180 of these orders shipped on time.

• What is the sample proportion of orders shipped on time?
• Does the sample data satisfy conditions necessary for the sample proportion to follow an approximately normal distribution?
• What is the mean and standard error of the sample distribution?
• If JCrew really ships 95% of its orders on time, what is probability that the proportion in a random sample of 200 orders is as small or smaller as the proportion in the audit?

Do problem 7.26 (p. 267 in the textbook).

Do problem 7.27 (p. 267 in the textbook).

A Gallup poll found that for those Americans who have lost weight, 31% believed the most effective strategy involved exercise. What is the probability that from random sample of 300 Americans the sample proportion falls between 29% and 33%?