Example Random samples of size n =2 are drawn from a finite population that consists of the numbers 2, 4, 6 and 8 without replacement. a-) Calculate the mean and the standard deviation of this population b-) List six possible random samples of size n=2 that can be drawn from this population and calculate their means. c-) Use the results in b-) to construct the sampling distribution of the mean. d-) Calculate the standard deviation of the sampling distribution.

Example 2: Tuition Cost The mean tuition cost at state universities throughout the USA is 4,260 USD per year (2002 year figures). Use this value as the population mean and assume that the population standard deviation is 900 USD. Suppose that a random sample of 50 state universities will be selected. A-) Show the sampling distribution of x̄ (where x̄ is the sample mean tuition cost for the 50 state universities) B-) What is the probability that the random sample will provide a sample mean within 250 USD of the population mean? C-) What is the probability that the simple random sample will provide a sample mean within 100 USD of the population mean?

Example 5: Ping-Pong Balls The diameter of a brand of Ping-Pong balls is approximately normally distributed, with a mean of 1.30 inches and a standard deviation of 0.04 inch. If you select a random sample of 16 Ping-Pong balls, A-) What is the sampling distribution of the sample mean? B-) What is the probability that sample mean is less than 1.28 inches? C-) What is the probability that sample mean is between 1.31 and 1.33 inches? D-) The probability is 60% that sample mean will be between what two values, symmetrically distributed around the population mean?

Example 6: E-Mails Time spent using e-mail per session is normally distributed, with a mean of 8 minutes and a standard deviation of 2 minutes. If you select a random sample of 25 sessions, A-) What is the probability that sample mean is between 7.8 and 8.2 minutes? B-) What is the probability that sample mean is between 7.5 and 8.0 minutes? C-) If you select a random sample of 100 sessions, what is the probability that sample mean is between 7.8 and 8.2 minutes? D-) Explain the difference in the results of (A) and (C).

Types of Survey Errors Coverage error Non response error Sampling error Measurement error Excluded from frame Follow up on nonresponses Random differences from sample to sample Bad or leading question

Z X Population Distribution Sampling Distribution Standard Normal Distribution ? ? ? ? ? ? ? ? ? ? Sample Standardize ? ? X Z

Sampling Distribution Properties As n increases, decreases Larger sample size Smaller sample size

Sampling Distribution Properties Normal Population Distribution (i.e. is unbiased ) Normal Sampling Distribution (has the same mean) Variation:

How Large is Large Enough? For most distributions, n ≥ 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n ≥ 15 For normal population distributions, the sampling distribution of the mean is always normally distributed