Sampling Distribution of a Sample Mean Lecture 30 Section 8.4 Tue, Mar 15, 2005

The Central Limit Theorem Begin with a population that has mean  and standard deviation . For sample size n, the sampling distribution of the sample mean is approximately normal with

The Central Limit Theorem The approximation gets better and better as the sample size gets larger and larger. For many populations, the distribution is almost exactly normal when n  10. For almost all populations, if n  30, then the distribution is almost exactly normal.

The Central Limit Theorem Special Case: If the original population is exactly normal, then the sampling distribution of the sample mean is exactly normal for any sample size. This is all summarized on page 500.

Example The population {1, 2, 3} has Mean 2. Variance 2/3. Standard deviation (2/3) = 0.8165.

Example When n = 3, the sample mean is (very) approximately normal with Mean 2. Standard deviation 0.8165/3 = 0.4714.

Example When n = 30, the sample mean is approximately (almost exactly) normal with Mean 2. Standard deviation 0.8165/30 = 0.1491.

Example If I collect, with replacement, a sample of 30 values from this population, what is the probability that my sample mean will be at least 2.2?

Let’s Do It! Let’s do it! 8.9, p. 502 – Probability of Accepting the Shipment. Find P(X > 250). Note that P(X < 250) is. Let’s do it! 8.11, p. 504 – Testing Hypotheses about the Mean Weight of Nuts. Find P(X < 15.8), i.e., the p-value. Let’s do it! 8.10, p. 503 –Mean Grocery Expenditure.

Estimating the Population Mean Example 8.12, p. 504 – Estimating the Population Mean Grocery Expenditure. The sampling distribution ofx is approximately normal with x = $60. x = $35/100 = $3.50. Based on the Empirical Rule, 95% of all samples have a mean within $7.00 of $60, that is, between $53 and $67.

Estimating a Population Proportion See the article “Water on airlines often unacceptable, finds EPA.” They found that the water in 20 out of 158 airliners contained coliform. So p^ = 20/158 = 0.1266 = 12.66%. What is a good estimate of p, the planes whose water contains coliform as a proportion of all planes?

Estimating a Population Proportion Based on our theory, there is a 95% chance that p^ is within 2 standard deviations of p. Therefore, there is a 95% chance that p is within 2 standard deviations of p^. That is, there’s a 95% chance that p is between p^ – 2p^ and p^ + 2p^.

Estimating a Population Proportion Compute p^ = 0.0265 = 2.65%. Therefore, we are 95% sure that the true proportion is between 7.36% and 17.96%. This is called a 95% confidence interval. How clean are municipal water systems? Based on the data, is it reasonable to believe that the water on airliners is in fact cleaner than the water in municipal water systems?