Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4Statistical InferenceStatistical InferenceExample 1Example 1Example 2Example 2Example 3Example 3Sample DataSlide 14Sampling DistributionSampling DistributionsSampling Distribution of the Sample Mean XAssumptionsAssumptionsCentral Limit TheoremCLT for ProportionsCentral Limit Theorem for MeansSampling Distribution of XSampling Distribution of XSampling Distribution of XSampling Distribution of XSampling Distribution of XExample 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 78/65Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 79/66Example 80/67Example 80/67Example 80/67Example 80/67Example 80/67Example 80/67Example 80/67Example 80/67Example 80/67Example 80/67ProbabilityProbabilityProbabilityExample 81/68Example 81/68Example 81/68Example 81/68Example 82/69Example 82/69Example 82/69Example 82/69Example 83/70Example 83/70Motivating ExampleMotivating Example SolutionSlide 79STAT 210Lecture 31Sampling Distribution of the Sample Mean XNovember 9, 2016Practice ProblemsSailboat: Pages 220 through 230Relevant problems: VIII.1 through VIII.6Recommended problems: VIII.5 and VIII.6Hummingbird: Pages 250 through 260Relevant problems: IX.1 through IX.6Recommended problems: IX.5 and IX.6Additional Reading and ExamplesSailboat: Read pages 213 through 218Pay particular attention to page 213Hummingbird: Read pages 243 through 248Pay particular attention to page 243ClickerStatistical inference involves using statistics computed from sample data to make statements about unknown population parameters.In this chapter the parameter of interest is the population mean m. We are assuming that data for the entire population is not known, and hence m is unknown, so statistical inferences in the form of confidence intervals and tests of significance will need to be performed.Statistical InferenceStatistical InferenceStatistical inference involves using statistics computed from sample data to make statements about unknown population parameters.When making statistical inferences, the first step is to identify the population of interest and the specific parameter of interest. Consider the following three examples.Example 1Of interest is to estimate the mean age of all students at this university.Example 1Of interest is to estimate the mean age of all students at this university. In this situation the population is all students at this university, and the parameter of interest is m = the mean age of all students at this university.Example 2It is conjectured that the mean cost of all women’s swimwear purchased in 2009 was $60, and of interest is to test this conjecture versus the alternative that the mean cost of all women’s swimwear purchased in 2009 was actually greater than $60.Example 2It is conjectured that the mean cost of all women’s swimwear purchased in 2009 was $60, and of interest is to test this conjecture versus the alternative that the mean cost of all women’s swimwear purchased in 2009 was actually greater than $60. In this situation the population consists of all women’s swimwear purchased in 2009, and the parameter of interest is m = the mean cost of all women’s swimwear purchased in 2009.Example 3General Motors advertises that the 2010 Chevy Equinox averages 32 miles per gallon when driven on the highway. Of interest is to test this claim versus the alternative that the mean miles per gallon when driven on the highway for all 2010 Chevy Equinox vehicles is different from 32 miles per gallon.Example 3General Motors advertises that the 2010 Chevy Equinox averages 32 miles per gallon when driven on the highway. Of interest is to test this claim versus the alternative that the mean miles per gallon when driven on the highway for all 2010 Chevy Equinox vehicles is different from 32 miles per gallon. In this situation the population consists of all 2010 Chevy Equinox vehicles driven on the highway, and the parameter of interest is m = the mean miles per gallon of all 2010 Chevy Equinox vehicles when driven on the highway.Sample DataOnce the population and parameter of interest are determined, a sample is selected from the population and data collected on the characteristic of interest for the individuals in the sample.From the sample data we compute the sample mean X, and the sample mean X becomes the basis for the inferences that will be made about the unknown population mean m. The sample mean X is called the point estimate of the population mean m.ClickerSampling DistributionA sampling distribution of a statistic is the distribution of values taken by the statistic in a large number of simple random samples of the same size n taken from the same population.Sampling DistributionsTo theoretically describe a sampling distributionwe must describe the (1) shape, (2) center, (3) spread, and (4) unusual features of the distribution.Suppose we know the population mean m and the population standard deviation s.Sampling Distribution of the Sample Mean XAssumptions1. The data being used to make inferences must be a simple random sample taken from the population.Assumptions1. The data being used to make inferences must be a simple random sample taken from the population.2. The population distribution must be known to be normal, or the sample size must be “large enough” for the Central Limit Theorem to apply.Central Limit TheoremThe Central Limit Theorem is a mathematical property that states that regardless of the shape of the original population, if the sample size is “large enough” then the shape of the sampling distribution will be approximately normal.CLT for ProportionsSince the population is not normal, we must use the Central Limit Theorem. If the sample size is “large enough” then the distribution of the sample proportion p is approximately normal.The sample will be “large enough” if BOTH the expected number of successes np and the expected number of failures n(1 - p) are greater than or equal to 10.Central Limit Theorem for MeansWhat is “large enough”?If the sample size is less than 15, then the original population must be normal.If the sample size is at least 15, then the Central Limit Theorem will apply unless the distribution is heavily skewed due to outliers (“at least 15” means
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