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AP Statistics Section 9.3B The Central Limit Theorem

AP Statistics Section 9.3B The Central Limit Theorem. In Section 9.3A, we saw that if we draw an SRS of size n from a population with a Normal distribution, , then the sample mean, , has a Normal distribution N (___, _____).

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AP Statistics Section 9.3B The Central Limit Theorem

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  1. AP Statistics Section 9.3BThe Central Limit Theorem

  2. In Section 9.3A, we saw that if we draw an SRS of size n from a population with a Normal distribution, , then the sample mean, , has a Normaldistribution N(___, _____)

  3. Although many populations have roughly Normal distributions, very few are exactly Normal. So what happens to when the population distribution is not Normal? In Activity 9B, the distribution of the ages of pennies should have been right- skewed, but as the sample size increased from 1 to 5 to 10 and then to 25, the distribution should have gotten closer and closer to a Normal distribution.

  4. This is true no matter what shape the population distribution has, as long as the population has a finite standard deviation . This famous fact of probability is called the central limit theorem.

  5. Central Limit TheoremDraw an SRS of size n from any population whatsoever with mean and standard deviation . When n is large, the sampling distribution of the sample mean is close to the Normal distribution .

  6. There are 3 situations to consider when discussing the shape of the sampling distribution of .

  7. 1. If the population has a Normal distribution, then the shape of the sampling distribution is Normal, regardless of the sample size.

  8. 2. If the population has any shape and the sample size is small, then the shape of the sampling distribution is similar to the shape of the parent population.

  9. 3. If the population has any shape and the sample size is large, then the shape of the sampling distribution is approximately Normal.

  10. **How large a sample size is needed for to be close to Normal? The farther the shape of the population is from Normal, the more observations are required.

  11. Example: The time a technician requires to perform preventative maintenance on an air-conditioning unit is an exponential distribution with the mean time hour and the standard deviation hour. Your company has a contract to maintain 70 of these units in an apartment building. You must schedule technicians’ times for a visit. Is it safe to budget an average of 1.1 hours for each unit? Or should you budget an average of 1.25 hours?

  12. The figure below summarizes the sampling distribution of . It reminds us of the big idea of a sampling distribution. Keep taking random samples of size n from a population with mean . Find the sample mean for each sample. Collect all the and display their distribution. That’s the sampling distribution of . Sampling distributions are the key to understanding statistical inference.

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