Sampling Distribution of a Sample Mean Lecture 30 Section 8.4 Mon, Mar 19, 2007.

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Presentation transcript:

Sampling Distribution of a Sample Mean Lecture 30 Section 8.4 Mon, Mar 19, 2007

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. That is, the sampling distribution “morphs” from the distribution of the original population to the normal distribution.

The Central Limit Theorem 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 Also, 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 pages 536 – 537.

Excel Assignment 2 In Excel Assignment 2, each quiz had a distribution of U(50, 100).   = 75,   = When we averaged 10 quiz scores at a time, what distribution did we get? QuizScores.xls. QuizScores.xls

Bag A vs. Bag B Bag A Bag B

Bag A vs. Bag B Use the TI-83 to compute the mean and standard of each population (Bag A and Bag B).

Bag A vs. Bag B The Bag A population:   =   = The Bag B population:   =   = 14.53

Bag A vs. Bag B The hypotheses:  H 0 : The bag is Bag A.  H 1 : The bag is Bag B. Suppose that we sample 10 vouchers (one at a time, with replacement). If the average value of the 10 vouchers is less than 35, we will accept H 0.

Bag A vs. Bag B What is  ?  Find the sampling distribution of  x if H 0 is true. What is  ?  Find the sampling distribution of  x if H 1 is true. How reliable is this test?

Bag A vs. Bag B Sampling distributions when n = 5: 35

Bag A vs. Bag B Sampling distributions when n = 5: 35

Example Suppose a brand of light bulb has a mean life of 750 hours with a standard deviation of 120 hours. What is the probability that 36 of these light bulbs would last a total of at least hours?