# VCU STAT 210 - Lecture24 (54 pages)

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## Lecture24

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- Pages:
- 54
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 24 Introduction to Confidence Intervals October 23 2017 Practice Problems Pages 187 and 188 Relevant problems VII 1 VII 2 VII 3 and VII 4 Recommended problems VII 1 VII 2 VII 3 VII 4 Additional Reading and Examples Read pages 185 and 186 Top Hat 2 Inference Statistical inference involves using statistics computed from data collected in a sample to make statements inferences about unknown population parameters Two types of statistical inference are estimation of parameters using confidence intervals and statistical tests about parameters In this chapter we learn the basic concepts associated with confidence intervals and with statistical tests Inference The first step in any inference procedure is to state the practical question that needs to be answered This involves specifying the population of interest and then the specific parameter that inferences need to be made about Motivating Example Suppose the population is all students at this university There are two parameters p the proportion of all students at this university who have children m mean IQ of all students at this university Motivating Example Suppose the population is all students at this university There are two parameters p the proportion of all students at this university who have children m mean IQ of all students at this university Does anyone know the proportion of all students at this university who have children Motivating Example Suppose the population is all students at this university There are two parameters p the proportion of all students at this university who have children m mean IQ of all students at this university Does anyone know the mean IQ of all students at this university Motivating Example Suppose the population is all students at this university There are two parameters p the proportion of all students at this university who have children m mean IQ of all students at this university Since data for all students is not known it is likely not possible to determine the value of either parameter This is when statistical inference comes into action Motivating Example Suppose the population is all students at this university If the parameter is p the proportion of all students at this university who have children and if data for all students is not available what could we do Motivating Example Suppose the population is all students at this university If the parameter is p the proportion of all students at this university who have children and if data for all students is not available what could we do 1 Select a sample from the population 2 Collect data for students in the sample 3 Compute the proportion of the students in the sample that have children Being from the sample this is a statistic Top Hat Motivating Example Suppose the population is all students at this university The parameter is p the proportion of all students at this university who have children and is unknown From our sample of students have children So the proportion of the sample who have children is This is an estimate of the parameter Motivating Example Suppose the population is all students at this university The parameter is p the proportion of all students at this university who have children and is unknown From our sample of students have children So the proportion of the sample who have children is Do you think the proportion of all students with children is exactly Motivating Example Suppose the population is all students at this university The parameter is p the proportion of all students at this university who have children and is unknown From our sample of students have children So the proportion of the sample who have children is Do you think the proportion of all students with children is exactly Most likely no and the statistical inference procedure that handles this situation is called a confidence interval Confidence Intervals Confidence intervals are statistical procedures that allow for the estimation of unknown population parameters The procedure involves both the calculation of the interval and then the interpretation of the interval The interpretation is the statistical inference Confidence Intervals To estimate an unknown population parameter we begin by selecting a sample from the population Once the sample is selected we then collect the necessary information from those in the sample The data collected from the sample is used to compute a statistic and this statistic becomes the starting point for the confidence interval and hence the statistical inference Confidence Intervals The data collected from the sample is used to compute a statistic and this statistic becomes the starting point for the confidence interval and hence the statistical inference The value computed from the sample data collected is referred to as the point estimate of the unknown population parameter Motivating Example Suppose the population is all students at this university The parameter is p the proportion of all students at this university who have children and is unknown From our sample of students have children So the proportion of the sample who have children is This sample proportion p is the point estimate of p the proportion of all students at the university with children Confidence Intervals If the sample is representative of the population then one would expect that the point estimate will be a good estimate of the population parameter and hence would be very close to the actual but unknown value of the parameter However it is very unlikely that the actual value of the population parameter will equal to the point estimate value exactly Confidence Intervals Therefore since the point estimate should be close to the population parameter but will likely not equal to it exactly to the point estimate we subtract and add a quantity called a margin of error to create an interval of values in which we hope the parameter will be contained This interval of values generated by subtracting and adding the margin of error is called a confidence interval In future chapters we will have a formula for the margin of error Confidence Intervals Therefore since the point estimate should be close to the population parameter but will likely not equal to it exactly to the point estimate we subtract and add a quantity called a margin of error to create an interval of values in which we hope the parameter will be contained The margin of error is considered a practical upper bound for the distance between the point estimate and the

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