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HW-pg. 596 (9.32 - 9.34) Ch. 9 Test FRIDAY 1-24-14 www.westex.org HS, Teacher Website 1-21-14 Warm up—AP Stats None! Name _________________________ AP Stats 9 Sampling Distributions 9.3 Sample Means Date _______ Objectives Given the mean and standard deviation of a population, calculate the mean and standard deviation for the sampling distribution of a sample mean. Identify the shape of the sampling distribution of a sample mean drawn from a population that has a Normal distribution. Recap Sample proportions come up when we are looking at _______________ variables. What proportion of U.S. adults have watched Survivor? What proportion of U.S. adults believe in God? Sample Means When we record _______________ variables such as income of a household, heart rate of a person, score on the math section of the SAT, we are interested in other statistics such as the median or _______ or ________________ _______________ of the variable. 9.3 describes the sampling distribution of the _______ of the responses in an _____. Looking at the histogram above tells us that: Means of random samples are _____ variable than individual observations. A more detailed examination of the distributions would point to a second principle: Means of random samples are _______ Normal than individual observations. For these 2 reasons sample means are popular in statistical inference. The Mean and the Standard Deviation of ___. The sampling distribution of ___ is the distribution of the values of ___ in all possible samples of the same size from the population. The first histogram above shows the distribution of a _______________, with mean µ = -3.5% while the second histogram shows the distribution of the __________ _______ ___ from all samples of size n = 5 from the population. The mean of all the values of ___ is _______, but the values of ___ are less spread out than the individual values in the ________________. Mean and Standard Deviation of a Sample Mean Suppose that ___ is the mean of an ___ of size __ drawn from a _______ population with mean µ and standard deviation σ. Then the mean of the sampling distribution of ___ is ___ = µ and its standard deviation is ___ = . The behavior of ___ in repeated samples is similar to the sample proportion ___: The sample mean ___ is an unbiased estimator of the population mean µ. The values of ___ are _____ spread out for larger samples. Their standard deviation decreases at the rate ___, so you must take a sample 4 times as large to cut the sd of __ in _______. You should use __________ for the sd of ___ only when the population is at least ___ times as large as the sample. (almost always the case) ***These facts about the mean and sd of ___ are true no matter what the _______________ _______________ looks like.*** (More to come about this tomorrow, how exciting!!!) Example 9.10 The heights of young women varies approximately according to the N(64.5, 2.5) distribution. If we choose 1 woman at random, the heights we get in repeated choices follow this distribution. ***The distribution of the population is also the distribution of ONE observation chosen at random.*** Now measure the height of an SRS of 10 young women. The sampling distribution of their sample mean height ___ will have mean ___ = ___ = 64.5 inches and sd This just reminds us that the heights of individual women vary widely about the population mean, whereas the average height of a sample of 10 women is _____ variable. Sampling Distribution of a Sample Mean from a Normal Population Draw an SRS of size n from a population that has the Normal distribution with mean µ and sd σ. Then the sample mean ___ has the Normal distribution with mean µ and sd . **Tomorrow we will consider the shape of the sampling distribution of ___ if the shape of the population is unknown or known to be non-Normal.** Example 9.11 a) What is the probability that a randomly selected young woman is taller than 66.5 inches? b) What is the probability that the mean height of an SRS of 10 young women is greater than 66.5 inches?