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Section 8.1 – Sampling Distributions and Estimators Sampling Distribution of a Statistic – A probability distribution for all possible values of the statistic computed from a sample of size n. Sampling Distribution of the Sample Mean – A probability distribution for all possible values of the sample mean computed from a sample of size n from a population with mean µ and standard deviation . We will perform our sampling with replacement, because this will result in independent events. The value of a statistic depends on the particular values included in the sample, and it generally varies from sample to sample, so we call this Sampling Variability. Example: The assets (in billions of dollars) of the three wealthiest people in the US are: 62 (Warren Buffet), 58 (Bill Gates), and 26 (Sheldon Adelson). Assume that samples of size 2 are randomly selected with replacement from this population of three values. a) List the different possible samples, calculate the mean of each of sample, and find the probability of each sample mean (ie, create the sampling distribution of the sample means) Sample Mean of the sample Probability b) Calculate the mean of the sampling distribution and the mean of the population. c) Calculate the standard deviation of the sampling distribution and the standard deviation of the population. 1 The Mean and Standard Deviation of the Sampling Distribution of x : Suppose that a simple random sample of size n is drawn from a large population with mean µ and standard deviation . The sampling distribution of x will have mean x and standard deviation x n . Note: •If the original population is normally distributed, then the distribution of the sample means will be normally distributed, regardless of the sample size, n. •If the original population is not normally distributed, then the sample means will be normally distributed as long as the sample size is large enough (usually n ≥ 30). (see pg. 434) The Central Limit Theorem: Regardless of the shape of the underlying population, the sampling distribution of the sample mean becomes approximately normal as the sample size n increases. For the applications in this section, •When working with an individual value from a normally distributed population, use z •When working with the mean of a sample (group), use z x x n Example: Scores for men on the verbal section of the SAT test are normally distributed with a mean of 509 and a standard deviation of 112. a) If one man is chosen, find the probability that he scores less than 590. b) If 16 men are chosen, find the probability that their mean score is less than 590. c) In part b, why can the central limit theorem be used even though the sample size is 16? 2 Example: For women aged 18-24, the systolic blood pressures are normally distributed with mean 114.8 and standard deviation 13.1. a) If you select one woman, find the probability that her systolic blood pressure is greater than 120. b) If you select 36 women, find the probability that their mean systolic blood pressure is greater than 120. 3

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