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Chapter 9 Day 4

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Warm - Up Harley Davidson motorcycles make up 14% of all motorcycles registered in the U.S. You plan to interview an SRS of 500 motorcycle owners. What is the approximate distribution of your sample that own Harleys? Standard deviation? Why can you “do this”? (use Rules of Thumb) How likely is your sample to contain 20% or more who own Harleys? How likely is your sample to contain at least 15% who own Harleys?

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**Mean and Standard Deviation of a Sample Mean**

Suppose that is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the mean of the sampling distribution of is μ and its standard deviation is σ/√n

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**The behavior of in repeated samples is much like that of the sample proportion**

The sample mean is an unbiased estimator of the population mean μ. The values of are less spread out for larger samples. Their standard deviation decreases at the rate √n, so you must take a samples 4 times as large to cut the standard deviation of in half. You should only use the recipe for standard deviation when the population is at least 10 times as large as the sample.

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Example The height of young women varies approximately according to the N(64.5,2.5) distribution. If we choose an SRS of 10 young women, find the mean and standard deviation of the sample.

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Central Limit Theorem Draw an SRS of size n from any population whatsoever with mean μ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean is close to the normal distribution N(μ,σ/√n) with mean μ and standard deviation σ/√n In other words, as sample size increases the distribution becomes more normal.

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Example A company that owns a fleet of cars for its sales force has found that the service lifetime of disc brake pads varies form car to car according to a normal distribution with mean μ= 55,000 and standard deviation σ = miles. The company installs a new brand of brake pads on 8 cars.

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If the new brand has the same lifetime distribution as the previous type, what is the distribution of the sample mean lifetime for the 8 cars? The average life of the pads on these 8 cars turns out to be = 51,800 miles. What is the probability that the sample mean lifetime is 51,800 miles or less if the lifetime distribution is unchanged? The company takes this probability as evidence that the average lifetime of the new brand of pads is less than 55,000 miles.

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**Remember the Law of Large Numbers?**

Draw observations at random from any population with finite mean μ. As the number of observations drawn increases, the mean of the observed values gets closer and closer to μ.

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