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Sampling Distribution of Sample Mean The sampling distribution of the Sample Mean, X, is the distribution of the sample mean values for all possible samples of the same size from the same population. If the parent population IS N(μ, σ), then for any sample size (small or large), the distribution: Central Limit Theorem (CLT) The sampling distribution of the Sample Mean, , is the distribution of the sample mean values for X all possible samples of the same size from the same population. If the parent population IS NOT N(μ, σ), then for n>30 the distribution is approximately: Example: Research on eating disorder Bulimia is an illness in which a person binges on food or has regular episodes of significant overeating and feels a loss of control. Usually is observed in young females. Research on eating disorder During the American Statistician (May 2001) study of female students who suffer from bulimia, each student completed a questionnaire from which a “fear of negative evaluation” (FNE) score was produced. Suppose the FNE scores of bulimic students have a distribution with mean 18 and standard deviation of 5. What is the probability that a randomly selected bulimic female has a greater than 15? Research on eating disorder During the American Statistician (May 2001) study of female students who suffer from bulimia, each student completed a questionnaire from which a “fear of negative evaluation” (FNE) score was produced. Suppose the FNE scores of bulimic students have a distribution with mean 18 and standard deviation of 5. What is the probability that the sample mean FNE score (of 100) is greater than 15? Research on eating disorder During the American Statistician (May 2001) study of female students who suffer from bulimia, each student completed a questionnaire from which a “fear of negative evaluation” (FNE) score was produced. Suppose the FNE scores of bulimic students have a distribution with mean 18 and standard deviation of 5. What is the probability that the sample mean FNE score (of 25) is greater than 15? Example: Body fat of American Men The percentage of fat (not the same as body mass index) in the bodies of American men is an approx. normal with mean of 15% and std of 5%. If these values were used to describe the body fat of U.S. Army men, then a measure of 20% or more body fat would characterize a soldier as obese. Body fat of American Men The percentage of fat of American men is an approx. normal (mean=15, std =5) Measure of 20% or more body fat would characterize a U.S. Army soldier as obese. What is the probability that a randomly chosen soldier would be considered obese, i.e., the percentage of body fat of a randomly chosen soldier in the U.S. Army is 20% or higher? Body fat of American Men The percentage of fat of American men is an approx. normal (mean=15, std =5) Measure of 20% or more body fat would characterize a U.S. Army soldier as obese. What is the probability that an average percentage of body fat of a randomly chosen 100 soldiers would be 20% or higher? Body fat of American Men The percentage of fat of American men is an approx. normal (mean=15, std =5) Measure of 20% or more body fat would characterize a U.S. Army soldier as obese. Given that the distribution of the percentage of the body fat of U.S. Army soldiers has the mean of 13% and the standard deviation of 3% (not necessarily normal), what is the probability that an average percentage of body fat of a randomly chosen 100 soldiers would be 20% or higher? Confidence Interval Estimation Confidence Interval estimates a range of values for the population parameter with a pre-defined level of confidence (e.g., 95% confidence interval). A value not in a CI can be rejected as possible value of the population parameter A value in a CI is an “acceptable” or “reasonable” possibility for the value of a population parameter Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). Using the Empirical Rule, define the lower and the upper bounds of the interval within which falls 95% of the most typical data values. Empirical Rule Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). The lower and the upper bounds of the interval within which falls 95% of the most typical data values. So we expect that 95% of the times patients will have to wait at ER during weekends 16 to 54 minutes. Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). Describe the sampling distribution of a sample average waiting time based on a random sample of n=100 patients who came during the last weekend. Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is a ~ Normal(μ=35 minutes, σ=9.5 minutes). Describe the sampling distribution of a sample average waiting time based on a random sample of n=100 patients who came during the last weekend. Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is a ~ Normal(μ=35 minutes, σ=9.5 minutes). Describe the sampling distribution of a sample average waiting time based on a random sample of n=100 patients who came during the last weekend. Using CLT (or the Sampling distribution rule) for Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is a ~ Normal(μ=35 minutes, σ=9.5 minutes). Using the Empirical Rule, identify the lower and the upper bounds of the interval within which falls 95% of the most typical values for a sample mean (in a random sample of 100 patients). Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is a ~ Normal(μ=35 minutes, σ=9.5 minutes). , we expect that 95% of the times patients will have to wait The loweratand upperweekends bounds of the interval within which ERthe during 33.1 to on average. falls 95%36.9 of theminutes, most typical values for a sample mean (in a random sample of 100 patients). We expect that 95% of the times patients will have to wait at ER during weekends 33.1 to 36.9 minutes, on average. Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is a ~ Normal(μ=35 minutes, σ=9.5 minutes). Based on two independent random samples of 100 patients we computed two values for sample mean 37.85 minutes and 33.75 minutes. Which one is better? Why? Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is a ~ Normal(μ=35 minutes, σ=9.5 minutes). Now assume that true population μ is unknown. How can we estimate μ? Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is a ~ Normal(μ=35 minutes, σ=9.5 minutes). But if the true population mean is unknown, then there is no way to say which estimate is better (more accurate). In terms of the problem, people would like to believe that the average waiting time in ER on weekends is actually closer to 33.75 minutes than 37.85 minutes. Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). Assume that the true population mean is unknown, but the std is known σ=9.5. With a random sample of 100 patients, we estimated the sample mean 33.75 minutes. What would a rough guess for the lower and the upper bounds of the interval within which the true population mean falls with the 95% of confidence? Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). In the part c) we found that 95% of the times patients have to wait at ER during weekends 33.1 to 36.9 minutes, on average, or or Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). We can show that Is the same as Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). Then our approximate 95% confidence interval for the true population mean μ is: Does the 95% Confidence Interval contain the true population mean of 35? Example: Estimating Mean Waiting Time in the Emergency Room. Waiting time in the ER during weekends is Normal(μ=35 minutes, σ=9.5 minutes). The approximate 95% confidence interval for the true population mean μ (based on the sample mean of 37.85): Does the 95% Confidence Interval contain the true population mean of 35?