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9.3: Sample Means Warm Up = HW Pass With regard to a particular gene, the percentages of genotypes AA, Aa, and aa in a particular population are 60%, 30%, and 10%, respectively. Furthermore, the percentages of these genotypes that contract a certain disease are 1%, 5%, and 20%, respectively. If a person does contract the disease, what is the probability that the person is genotype AA? a) .006 b) .010 c) .041 d) .146 e) .600 Sample Proportions vs. Sample Means Sample proportions (Section 9.2) are most often used when we are interested in categorical variables. For example, “yes” or “no” questions. Sample means are most often used when we are interested in numerical or quantitative variables. For example, the average rate of return on stocks in the stock market. The distribution of returns for 1,815 New York Stock Exchange common stocks in 1987 The mean return for 1,815 stocks is μ = -3.5% and the distribution shows a very wide spread. Since each stock represents an individual sample, the sample size n = 1. The distribution of returns for all possible portfolios that invested equal amounts in A portfolio is just a sample 5 stocks each ofoffive stocks in 1987 and its return is the average return for the 5 stocks chosen. The mean return for all portfolios is still -3.5%, but the variation (spread) among portfolios is much less than the variation among individual stocks. The sample size is n = 5. Note that a larger sample size implies less variability. Life Lessons Learned from the Stock Market Statistical Inference is 1) Averages are less observations. the process of using samples answer variable thantoindividual specific questions about a population with a known degree of confidence. 2) Averages are more normal than individual observations. These two facts contribute to the popularity of sample means in statistical inference. Mean and Standard Deviation of a Sample Mean Suppose that x is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the mean of the sampling distribution of x is x The standard deviation is x n Sample Mean Basics The sample mean x-bar is an unbiased estimator of the population mean μ. The values of x-bar are less spread out for larger samples. You should only use the recipe for standard deviation of x-bar when the population is at least 10 times as large as the sample. These facts are true no matter what the shape of the population distribution. Ex 1: Young Women’s Heights The height of young women varies approximately according to the N(64.5, 2.5) distribution. We could safely say the if we repeatedly select one woman at random, the heights we get will also follow this distribution. But, what will happen if we begin choosing samples of 10 women at random? What will be the sample mean height x-bar of the sampling distribution? x 64.5 inches What about the standard deviation? 2.5 x 0.79 inch n 10 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 standard deviation σ. Then the sample mean x-bar has the normal distribution N( , / n ) Ex 2: More on Young Women’s Heights What is the probability that a randomly selected young woman is taller than 66.5 inches? What about the probability that the mean height of an SRS of 10 young women is greater than 66.5 inches? • Take a few minutes to try and answer these questions using normal calculations. (start with converting to z-scores) What is the probability that a randomly selected young woman is taller than 66.5 inches? 66.5 64.5 z 0.80 2.5 P( X 66.5) P( z 0.80) The probability of choosing a young woman at random whose height exceeds 66.5 inches is about 0.21. 1 0.7881 0.2119 What is the probability that the mean height of an SRS of 10 young women is greater than 66.5 inches? 66.5 64.5 z 2.53 0.79 P( x 66.5) P( z 2.53) It is very unlikely (< 1% chance) that we would draw an SRS of 10 young women whose average height exceeds 66.5 inches. 1 0.9943 0.0057 The sampling distribution of the mean height x-bar of 10 young women compared with the distribution of the height of a single woman chosen at random What have we learned? The average of n results (the sample mean xbar) is less variable than a single measurement. Question… Does x-bar still have a normal distribution even when the population distribution is not normal? Central Limit Theorem Draw an SRS of size n from any population whatsoever with mean μ and standard deviation σ. When n is large, the sampling distribution of the sample mean x-bar is very close to the Normal distribution N( , / n ) How large is large enough? How large a sample size n is needed for xbar to be close to Normal depends on the population distribution. More observations are required if the shape of the population distribution is far from Normal. Three scenarios to consider… 1) The population has a Normal distribution – shape of sampling distribution: Normal, regardless of sample size. 2) Any population shape, small n – shape of sampling distribution: similar to shape of the parent population. 3) Any population shape, large n – shape of sampling distribution: close to Normal (Central Limit Theorem) The sampling distribution is normal if the population distribution is normal. It will be approximately normal for large samples regardless of the shape of the population distribution.