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Warm Up

Using the Central Limit Theorem, calculate the probability that a teacher's students will achieve an average score of at least 3 on the AP Statistics exam.

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Warm Up

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  1. Warm Up • An AP Statistics teacher had 63 students preparing to take the AP exam. Though they were obviously not a random sample, he considered his students to be typical of all national students. The average score on the AP exam is a 2.859 with a standard deviation of 1.324. What’s the probability that his students will achieve an average score of at least 3?

  2. Ch. 18 – Sampling Distribution Models(Day 3 – The Central Limit Theorem) Part V – From the Data at Hand to the World at Large

  3. Normally Distributed Variables • Over the last few days, we have solved several problems involving normally distributed variables such as height, electric bills, test scores… • We were able to use the standard normal (z) distribution to solve these problems • What if you were asked to find the probability that the price of a randomly selected home fell in a certain range? Could you use z? • No – since home prices are likely to be skewed right, the normal distribution wouldn’t work!

  4. Sample Means (again) • It is the fact that there are a few extreme values on the right of the distribution of home prices that stops us from using the Normal model to represent it • However, what if we were examining samples of 20 houses, instead of just individual home prices – what would this distribution look like? • The effect of skew in a distribution is lessened when we look at sample means instead of the distribution of individual values

  5. The Central Limit Theorem • Central Limit Theorem: The mean of a random sample, even when the variable being measured is not normally distributed, has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better this approximation will be.

  6. In other words… • It doesn’t matter whether your original distribution was normal. The distribution of sample means will follow the Normal model anyway, as long as n is large enough. • How large is large enough? • This depends on the shape of the original distribution • The more “non-normal” the original distribution is, the larger n has to be

  7. Conditions Revisited • To use the Normal model to find probabilities involving sample means, the following conditions must be present: 1) Randomization: The sample must be selected randomly 2) 10% Condition: The sample size must be less than 10% of the population size 3) Normal Population or large sample size (n): • If x follows a normal distribution, the size of the sample doesn’t matter. • If x is not normally distributed, then n must be large enough to make the sampling distribution approximately normal. We will come back to this idea in a later chapter.

  8. Homework 18-3 • p. 432 # 29, 30, 49

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