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Sampling Variability and Sampling Distributions How can a sample be used to accurately present the characteristics of a population? Statistic is any quantity computed from values in a sample Sampling distribution the distribution of one statistic over many samples often look at histograms to determine patterns Sampling variability is the recognition that the value of a statistic depends on the sample selected , the size of the sample and how many samples are evaluated Since: x x x n since the average of the samples approximates the pop ave. Consider a population consisting of the following five values {8, 14, 16,10, 11}, which represent the number of video rentals during the academic year for each of the five housemates. a) compute the mean of this population b) select a random sample of size 2 by writing the numbers on slips of paper and selecting two. Compute the mean of the sample. c) repeatedly select samples of size two, and compute the associated values until you have looked at 25 samples d) construct a histogram using the 25 samples. Are most of the values near the population mean? Do they differ a lot from sample to sample or do they tend to be similar? Homework page 409-411 evens For #12 the info is on pages 406 and 409 If is based on a large # of items (n) in the sample the x is closer to µ than if n is small ( although x does approximate µ regardless of the size of n) If n is large σ is closer to Sx (meaning the data is less variable) x Rules of the Sampling Distribution 1) x 2) x (correct as long as no more than 5% of the population is included) n 3) when the population is normal, the sampling distribution of x is also normal for any sample size of n 4) When n is sufficiently large (ie approx 30+) the sampling distribution of x is approximated by a normal curve even when the population is not. (known as the CENTRAL LIMIT THEOREM) If n is large or the population distribution is normal z x x x x x n is the standard deviation of the sample averages Example 8.7 page 418 A soft-drink company claims that on average, cans contain 12 oz. of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with σ= .16 oz. Sixteen cans are to be selected and the soda volume determined for each one. Let x denote the resulting sample average soda volume. Because the x distribution is normal, the sampling distribution is normal. If the bottler’s claim is correct, the sample distribution of has a mean value 12 and a standard deviation of _ _ x n x .16 16 .04 a) determine the probability that a can of soda contains between 11.96 oz and 12.08 oz Z-chart b) if you choose a can of soda at random, what is the probability that the manufacturer is correct if your soda has at least 12.1 oz in it. HW Pg 420 to 422 8.14 to 8.22 even The Sample proportion p = number of successes RULES: 1) µp = π 2) σp = (1 ) # in the sample(n) n 3) when n is large and π is not too near to 0 or 1 the sampling distribution of p is approximately normal --the further π is from .5 the larger n must be for the sampling x distribution to be approx normal. Rule of thumb: conservative liberal If nπ ≥ 10 and n(1-π) ≥10 nπ≥5 and n(1-π)≥5 Note: the book tends to use the conservative method and the AP test tends to use the liberal method according to the AP listserve it doesn’t make a difference just show your set up!!!!! AP Statistics Example 8.11 page 426 The proportion of all blood recipients stricken with viral hepatitis was given as .07 in an article in the American Journal of medicine. Suppose a new treatment is believed to reduce the incidence rate of viral hepatitis. This treatment is given to 200 blood recipients. Only 6 of the 200 contracted hepatitis. It appears that the new treatment is better since p = 6/200 = .03. Determine if the sampling distribution of p is normally distributed Determine the probability that the sample distribution p could be less than .03 if the treatment were ineffective (i.e. it did not change --the original value of those contracting viral hepatitis under the old treatment) Hw pg 427 8.28 to 8.32 even Formulas: Distribution of x x x n If n is large the x dist is Normal even if the pop isn’t Distribution of p 1) µp = π 2) σp = (1 ) n The further π is from .5 the larger n must be for the distribution to be considered normal so: show the conservative or liberal test Homework pg 428 evens Test set up: 9 multiple choice—1 point each 3 computational—total of 45 points Test total 54 points