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Sampling Distributions Welcome to inference!!!! Central Limit Theorem Chapter 9.1/9.3 Day 2 Central Limit Theorem (CLT) As you take more and more SRS’s of the same size, the distribution of their means will get closer and closer to a normal curve centered around the true population mean…NO MATTER WHAT THE SHAPE OF THE PARENT POPULATION!! If you take large enough samples…(around 25 or 30) 1. The histogram of the samples will appear approximately normal (bell shaped). 2. The larger the sample size (n), the smaller the standard deviation will be and the more constricted the graph will be. 9.3 3 Possibilities for Sampling Distributions (a) The parent population is Normal The sampling distribution will be Normal, regardless of sample size (b) The sampling distribution will be similar to the shape of the parent population (not Normal) (c) Any population shape, small sample size Any population shape, large sample size The sampling distribution will be close to Normal (CLT) Example 1 The length of pregnancy from conception to birth varies has a mean of 266 days and a standard deviation of 16 days What is the probability that a woman chosen at random has a pregnancy lasting more than 270 days? CANNOT CALCULATE! The problem did not state that the population was Normal and the sample size of 1 is not large enough to perform a calculation. What is the mean and standard deviation of my sampling distribution? What is the probability that an SRS of 36 women have pregnancies averaging more than 270 days? Example 1 The length of pregnancy from conception to birth varies has a mean of 266 days and a standard deviation of 16 days What is the probability that an SRS of 36 women have pregnancies averaging more than 270 days? Check two conditions… (1) N≥10n? We can assume that there are more than 360 pregnant women in the population. (2) n≥25? The sample size of 36 women is large enough to assume that the sampling distribution is Normal based on the Central Limit Theorem. Do the calculations 16 ) 36 Normal CDF(270, 1E99, 266, The probability that an SRS of 36 women have pregnancies averaging more than 270 days is about 0.0668.