# VCU STAT 210 - Lecture26 (78 pages)

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## Lecture26

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- Pages:
- 78
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 26 Sampling Distributions of the Sample Proportion p October 27 2017 Practice Problems Sailboat Pages 252 through 257 Relevant problems IX 1 through IX 7 Recommended problems IX 1 IX 3 and IX 7 Hummingbird Pages 210 through 215 Relevant problems VIII 1 through VIII 7 Recommended problems VIII 1 VIII 3 and VIII 7 Additional Reading and Examples Sailboat Read pages 248 through 251 Pay particular attention to pages 248 249 Hummingbird Read pages 206 through 209 Pay particular attention to pages 206 207 Top Hat Motivating Example For 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 Inference Statistical 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 Inference Statistical 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 1 Of 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 1 Of 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 2 MTV 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 2 MTV 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 3 It 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 3 It 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 Data Once 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 Hat Sampling Distribution of the Sample Proportion p Consider 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 Variable We 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 Variable A 0 1 random variable has the following shape which obviously is not a normal distribution p 1 p 0 1 Sampling Distribution A 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 Distribution Example Suppose the population consists of all VCU students The parameter of interest is p the proportion of all VCU students who have children Do you know the proportion of all VCU students who have children Sampling Distribution Example Suppose the population consists of all VCU students The parameter of interest is p the proportion of all VCU students who have children Do you know the proportion of all VCU students who have children Assuming no then it must be estimated using a statistical inference procedure Sampling Distribution Example Suppose the population consists of all VCU students The parameter of interest is p the proportion of all VCU students who have children Suppose each person in this room selects a simple random sample of n 100 VCU students and computes p the proportion of the students in their sample that have children Sampling Distribution Example Suppose the population consists of all VCU students The parameter of interest is p the proportion of all VCU students who have children Student 1 sample of 100 students p 3

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