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Silent Do Now (5 minutes) *Take out your homework and find your partner!

Silent Do Now (5 minutes) *Take out your homework and find your partner!. 1. State police believe that 70% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 80 randomly selected cars.

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Silent Do Now (5 minutes) *Take out your homework and find your partner!

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  1. Silent Do Now (5 minutes)*Take out your homework and find your partner! 1. State police believe that 70% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 80 randomly selected cars. What is the mean and standard deviation of the sampling distribution for sample proportions? Can this distribution be approximated by a normal curve? (Hint: check assumption) What is the probability that the proportion of drivers that exceed the speed limit in this sample is less than 60%?

  2. Agenda Review Do Now/Homework Pass Back Quizzes Lesson on Sample Means Practice Problems

  3. The Central Limit Theorem • As you recall, the LAW OF LARGE NUMBERS says that with an increase in the number of TRIALS, our probability will get closer and closer to the TRUE probability or proportion. • In a similar fashion, the CENTRAL LIMIT THEOREM says that no matter what shape a population distribution has, as long as the population has a finite standard deviation, with an increased SAMPLE SIZE the distribution will always approach a NORMAL DISTRIBUTION. It will rarely be exactly normal, but it will always be roughly normal. ***** This is why we could approximate binomial probabilities using the normal distribution if “n” was large enough AND this is why we can approximate probabilities for samples using the normal distribution if “n” is large enough!*****

  4. Sampling Distributions with Means If we draw a SAMPLE of size n from a population that has the normal distribution with a mean and standard deviation of , then the sample mean has the Normal Distribution as well.

  5. Mean and Standard Deviation of Sample Means Mean for x-bar is Standard Deviation for x-bar is Please Note: Similar to sample proportions, we should only use the recipe for the standard deviation of only when the population is at least 10 times as large as the sample; that is when N greater than or equal to 10n.

  6. Example 1: Finding A Probability for a Sample Mean The height of women follows a normal distribution with a mean of 64.5 inches and a standard deviation of 2.5 inches. What is the probability that the mean height of a simple random sample of 10 young women is greater than 66.5 inches? Check Conditions (np is greater than or equal to 10, n(1-p) is greater than or equal to 10, N is greater than or equal to 10n) 2. If conditions are met, can apply the normal distribution! P(x-bar > 66.5) Sample mean = = 64.5 z = 66.5 – 64.5 Sample SD = = 2.5/square root of 10 = 0.79 0.79 z = 2.53 p = 1-.9943 p = .57 %

  7. Thinking Back! What if we wanted to know the probability that a randomly selected young woman is taller than 66.5 inches? How would we calculate this? How is this different?

  8. Guided Practice • The army reports that the distribution of head circumference among soldiers is approximately normal with mean 22.8 inches and standard deviation of 1.1 inches. • What is the probability that a randomly selected soldier’s head will have a circumference that is greater than 23.5 inches? • What is the probability that a random sample of five soldiers will have an average head circumference that is greater than 23.5 inches?

  9. Independent Practice Three Problems * If you finish early, start on your homework…

  10. Homework Pg. 595 #9.31, 9.32 Pg. 601 #9.36, 9.38, 9.39

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