Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4Motivating ExampleStatistical InferenceStatistical InferenceExample 1Example 1Example 2Example 2Example 3Example 3Sample DataChapter ComparisonsSlide 16Sampling DistributionSampling DistributionsSampling Distribution of the Sample Mean XAssumptionsAssumptionsShape of 0-1 Random VariableCentral Limit TheoremCLT for ProportionsSlide 25Central Limit Theorem for MeansSampling Distribution of XSampling Distribution of XSampling Distribution of XSampling Distribution of XSampling Distribution of XExample 65/78Example 65/78Example 65/78Example 65/78Example 65/78Example 65/78Example 65/78Example 65Example 65/78Example 65/78Example 65/78Example 65/78Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 66/79Example 67/80Example 67/80Example 67/80Example 67/80Example 67/80Example 67/80Example 67/80Example 67/80Example 67/80Example 67/80Motivating ExampleMotivating Example SolutionProbabilityProbabilityProbabilityExample 68/81Example 68/81Example 68/81Example 68/81Example 68/81Example 69/82Example 69/82Example 69/82Example 69/82Example 69/82Example 70/83Example 70/83STAT 210Lecture 31Sampling Distributions of the Sample Mean XNovember 8, 2017Practice 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 243Top Hat 2Motivating ExampleStudents often ask questions about the average score on the STAT 210 final exam and other related questions. We will return to this problem and describe the sampling distribution of the mean grade on the STAT 210 final exam for a simple random sample of 49 students.Statistical InferenceStatistical 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, and hence statistical inferences in the form of confidence intervals and statistical tests will need to be done.Statistical 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.Chapter Comparisons•Last chapter: PROPORTIONS–Population proportion p–Sample proportion p is the point estimateThis chapter: MEANS–Population mean m–Sample mean X is the point estimateTop HatSampling 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.For example, X = person’s age, m = 28.9, s = 4.8Sampling Distribution of the Sample Mean XAssumptions1. The data being used to make inferences must be a simple random sample from the population.Assumptions1. The data being used to make inferences must be a simple random sample 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.Shape of 0-1 Random VariableA 0-1 random variable has the following shape, which obviously is not a normal distribution. p 1- p 0 1Central 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”
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