1. Chapter 7: Introduction to Sampling Distributions Section 2: The Central Limit Theorem

2. Theorem 7.1  Let x be a random variable with a normal distribution whose mean is and standard deviation is. Let be the sample mean corresponding to random samples of size n taken from the x distribution. 1. The distribution is a normal distribution. 2. The mean of the distribution is. 3.The standard deviation of the distribution is.

3. To find the z value: z =

4. Example  Let x represent the length of a single trout taken at random from the pond. This is normal distribution with a mean of 10.2 inches and standard deviation of 1.4 inches. a.) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long? b.) What is the probability that the mean length of five trout taken at random is between 8 and 12 inches?

5. Central Limit Theorem IIIIf x possesses any distribution with mean and standard deviation, then the sample mean based on a random sample of size n will approach a normal distribution when n is sufficiently large (n > 30).

6.  Suppose we know that the x distribution has a mean = 30 and a standard deviation = 8, but we have no information as to whether or not the x distribution is normal. If we draw samples of size 35 from the x distribution and represents the sample mean, what can you say about the distribution?

7.  Suppose x has a normal distribution with mean = 20 and = 5. If we draw random samples of size 10 from the x distribution, and represents the sample mean, what can you say about the distribution?  Suppose you did not know that x had a normal distribution. Would you be justified in saying that the distribution is approximately normal if the sample size was n = 8?

8. 1. Suppose we have no information as to whether or not the x distribution is normal. If we draw samples of n = 5 from the x distribution and represents the sample mean, will the distribution be approximately normal? Explain why or why not. Explain why or why not.

9. 2. Suppose x has a normal distribution with mean = 12 and = 3. If we draw random samples of size 10 from the x distribution, and represents the sample mean, what can you say about the distribution? 2. Suppose x has a normal distribution with mean = 12 and = 3. If we draw random samples of size 10 from the x distribution, and represents the sample mean, what can you say about the distribution? 3. Suppose you do not know if x has a normal distribution. If we draw random samples of size 40, will the distribution be approximately normal? 3. Suppose you do not know if x has a normal distribution. If we draw random samples of size 40, will the distribution be approximately normal? Explain why or why not. Explain why or why not.