Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4Motivating ExampleStatistical InferenceStatistical InferenceExample 1Example 1Example 2Example 2Example 3Example 3Sample DataSlide 15Sampling Distribution of the Sample Proportion p0-1 Random VariableShape of 0-1 Random VariableSampling DistributionSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling DistributionsAssumptionsAssumptionsCentral Limit TheoremSampling Distribution of pSampling Distribution of pSampling Distribution of pSampling Distribution of pSampling Distribution of pExample 83/65Example 83/65Example 83/65Example 83/65Example 83/65Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 84/66Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Example 85/67Motivating ExampleMotivating Example SolutionMotivating Example SolutionProbabilityProbabilityProbabilityExample 86/68Example 87/69Example 87/69Example 87/69Example 87/69Example 88/70Example 88/70Example 88/70STAT 210Lecture 26Sampling Distributions of the Sample Proportion pOctober 27, 2017Practice ProblemsSailboat: Pages 252 through 257Relevant problems: IX.1 through IX.7Recommended problems: IX.1, IX.3 and IX.7Hummingbird: Pages 210 through 215Relevant problems: VIII.1 through VIII.7Recommended problems: VIII.1, VIII.3 and VIII.7Additional Reading and ExamplesSailboat: Read pages 248 through 251Pay particular attention to pages 248 – 249Hummingbird: Read pages 206 through 209Pay particular attention to pages 206 - 207Top HatMotivating ExampleFor both faculty and students, exercise is an activity that can lead to both a healthy body and a healthy mind. The goal is to describe the distribution of the proportion of a simple random sample of 225 VCU faculty and students who exercise at the VCU Cary Street Recreational Center for two hours or longer.Statistical InferenceStatistical inference involves using statistics computed from sample data to make statements about unknown population parameters.In this chapter we will learn how to use the sample proportion p to make statistical inferences about the population proportion p.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 proportion of all students at this university who have children. What is the population of interest?What is the parameter of interest?Example 1Of interest is to estimate the proportion of all students at this university who have children. In this situation the population of interest is all students at this university, and the parameter of interest is p = the proportion of all students at this university with children.Example 2MTV was launched on August 1, 1981 and initially played music videos guided by on-air hosts. Currently MTV plays a limited selection of music videos, but primarily broadcasts a variety of popular culture and reality television shows targeted at adolescents and young adults. However, since the show started in 1981 there is still a loyal following of viewers from the 1980’s, and of interest is to estimate the proportion of current viewers who are age 30 or older. What is the population of interest?What is the parameter of interest?Example 2MTV was launched on August 1, 1981 and initially played music videos guided by on-air hosts. Currently MTV plays a limited selection of music videos, but primarily broadcasts a variety of popular culture and reality television shows targeted at adolescents and young adults. However, since the show started in 1981 there is still a loyal following of viewers from the 1980’s, and of interest is to estimate the proportion of current viewers who are age 30 or older. In this situation the population consists of all current viewers of MTV, and the parameter of interest is p = the proportion of all current MTV viewers who are age 30 or older.Example 3It is conjectured that at any given Major League Baseball game approximately 5% of fans in attendance are attending their first Major League Baseball game. Of interest is to test this claim versus the alternative that the proportion of fans in attendance at baseball games who are attending their first Major League Game is different from 0.05. What is the population of interest?What is the parameter of interest?Example 3It is conjectured that at any given Major League Baseball game approximately 5% of fans in attendance are attending their first Major League Baseball game. Of interest is to test this claim versus the alternative that the proportion of fans in attendance at baseball games who are attending their first Major League Game is different from 0.05. In this situation the population consists of all fans attending Major League Baseball games, and the parameter of interest is p = the proportion of all fans attending Major League Baseball games who are attending their first Major League Baseball game.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 proportion p, and the sample proportion p becomes the basis for the inferences that will be made about the unknown population proportion p. The sample proportion p is called the point estimate of the population proportion p.Top HatSampling Distribution of the Sample Proportion pConsider a random variable X that can take only two possible values: a “success” or a “failure”.Examples: 1. A TV set either works or doesn’t work. 2. A student passes a test or fails a test. 3. The light is on or off.0-1 Random VariableWe code a “success” as a 1 and we code a “failure” as a 0.The proportion of “successes” in the population (of 1’s) is denoted p, and hence the proportion of “failures” (of 0’s) is 1 - p.This creates what is called a 0-1 random variable.Shape of 0-1 Random VariableA 0-1 random variable
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