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Warm-up 7.2 Generating Sampling Distributions Answers to Warm-up (C) X is close to the mean and so will have a z-score close to 0. Boxplots, if they show an isolated point, will show only outliers. X and the two clusters are clearly visible in a stemplot of these data. School of the day! Block 4 School of the day! Block 5 Answers to H.W. E #4 and 5 Answer #5 7.2 Generating Sampling Distributions Sampling Distribution of a Mean If you toss a fair die 10,000 what do you expect the histogram of the results to look like? How would you expect the histogram to look like if you were graphing the results of the average of two die? Central Limit Theorem The sampling distribution of any mean becomes more nearly Normal as the sample size grows. Most importantly the observations need to be independent and collected with randomization. FYI “central” in the theorem name means “fundamental” CLT and Equations • The CLT requires essentially the same assumptions and conditions from modeling proportions: Independence, Sample size, Randomization, 10% and large enough sample The standard deviation of the sampling distribution is sometimes called the standard error of the mean. x n standard dev. of pop. / sample size Properties of Sampling Distribution of Sample Mean pg 430 If a random sample of size n is selected from a population with mean and standard deviation , then • The mean x of the sampling distribution of x equation equals the mean of the population, : • The standard deviation x , the sampling distribution of x , sometimes called the standard error of the mean: x standard dev. of pop. / sample size n • Using the formula above you can find standard error of the sample mean without simulation Number of Children Problem Physical Education Department and BMI study A college physical education department asked a random sample of 200 female students to self-report their heights and weights, but the percentage of students with body mass indexes over 25 seemed suspiciously low. One possible explanation may be that the respondents “shaded“ their weights down a bit. The CDC reports that the mean weight of 18-year-old women is 143.74 lb, with a standard deviation of 51.54 lb, but these 200 randomly selected women reported a mean weight of only 140 lb. Question: Based on the Central Limit Theorem and the 68-95-99.7 Rule, does the mean weight in this sample seem exceptionally low or might this just be random sample-to-sample variation? Common Mistakes on Test what is the mean and standard deviation for the number of defects for pairs of shoes produced by this company. Form A 1. This table gives the percentage of women who ultimately have a given number of children. For example, 19% of women ultimately have 3 children. What is the probability that two randomly selected women will have a combined total of exactly 2 children? 0 and 2, 1 and 1, 2 and 0 0.18* 0.35 + 0.17*0.17+ 0.35* 0.18 = 0.159 Form B 2. For the sake of efficiency, a shoe company decides to produce the left shoe of each pair at one site and the right shoe at a different site. If the two sites produce shoes with a number of defects reflected by 1 0 . 002 , 1 0 . 15 and 2 0 . 005 , 2 0 . 18 , X Y X Y 0 . 002 0 . 005 0 . 007 X Y X Y 0 . 15 2 2 2 2 0 . 18 2 0 . 0549 . 234 Expected Number of Success and Expected Number of Trials Binomial Distribution: Flipping a coin 6 times, about how many times flip head on average. n p 6 ( 0 . 5 ) 3 x In a simple random sample of 15 students, how many are expected to be younger than 20 . 60% of students are under 20. Geometric Distribution: Flipping a coin, when do you expect 1 (on average) to have your first success. x p What is the expected number of interview before the second person without health insurance is found? (16% no health insurance) Expected Number of Success and Expected Number of Trials Binomial Distribution: Flipping a coin 6 times, about how many times flip head on average. n p 6 ( 0 . 5 ) 3 x What is the average number of students without laptops you would expect to find after sampling 5 random students? (60% w/ laptops) Geometric Distribution: Flipping a coin, when do you expect (on average) to have your first success. 1 x p On average how many otters would biologists have to check before finding an infected otter? (20% are infected)