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AP Statistics Section 7.3B The Central Limit Theorem In Section 7.3A, we saw that if we draw an SRS of size n from a population with a Normal distribution, N( , ), then the sample mean, x , has a Normal _____) distribution N(___, n Note : For x n , N 10n Although many populations have roughly Normal distributions, there are certainly some population distributions that are not Normal. So what happens to the shape of the distribution of x when the population distribution is not Normal? The website lock5stat.com/statkey/ is awesome for visualizing distributions. Notice the bar for sampling distributions, in particular the “mean”. If you click on the mean, the screen at the right appears. In the upper left hand corner there is a drop down menu with preloaded sets of data. Let’s use the salaries of baseball players in the year 2012. One you have clicked on the baseball salaries the distribution of the population will appear at the upper right along with the mean of the distribution and the standard deviation. a) Describe the shape of the population distribution. Severely right skewed b) Give the mean and standard deviation of the population distribution. 3.439 4.698 Now let’s look at the sampling distribution of x . Notice Statkey allows you to choose samples of different size – let’s start with samples of size 10 and generate 1 sample. What does this one dot on the distribution represent? The mean of a random sample of 10 baseball salaries Now let’s generate 1000 samples of size 10. a) Describe the shape of the sampling distribution. Slightly right skewed b) Give the mean and standard deviation of the sampling distribution. 3.512 1.517 Now let’s generate 1000 samples of size 30. a) Describe the shape of the sampling distribution. Bell shaped Approx.Normal b) Give the mean and standard deviation of the population distribution. 3.399 0.822 Central Limit Theorem Draw 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 N ( , ) . n There are 3 situations to consider when discussing the shape of the sampling distribution of x . 1. If the population has a Normal distribution, then the shape of the sampling distribution is Normal, regardless of the sample size. 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. 3. If the population has any shape and the sample size is large, then the shape of the sampling distribution is approximately Normal. **How large a sample size is needed x for to be close to Normal? The farther the shape of the population is from Normal, the more observations are required. Example: The time a technician requires to perform preventative maintenance on an air-conditioning unit is an exponential distribution with the mean time 1 hour and the standard deviation 1 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? Because 70 30, by the CLT, the dist. of x is approx. N(1, 1 70 ) Pop. of all such AC units 10(70 )or 700 P( x 1.1) .201 P( x 1.25) .018 At 1.1 hrs/call the tech will run late 20% of the time but at 1.25 hrs/call the tech will only run late 1.8% of the time The figure below summarizes the sampling distribution of x . 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 x for each sample. Collect all the x' s and display their distribution. That’s the sampling distribution of x . Sampling distributions are the key to understanding statistical inference. N 10n n The dist. is Normal if the pop. dist. is Normal. The dist. is approx. Normal for large samples in any case.