The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Chapter 6 Introduction to Inference Lecture 16 Estimation with Confidence 3 22 11 Lecture 16 1 Introduction Statistical Inference Sample Population Statistical inference is a class of procedures by which we acquire information about populations from samples Three procedures for making inferences Point estimation Confidence interval Hypotheses testing 3 22 11 Lecture 16 2 TV Watching Time The number and the types of television programs and commercials targeted at children are affected by the amount of time children watch TV Average time children watch TV A survey was conducted among 100 children in which they were asked to record the number of hours they watched TV per week The sample mean is x 27 191 3 22 11 Lecture 16 3 Point Estimation Use sample mean to estimate population mean A point estimator makes inference about a population by estimating the value of an unknown parameter using a single numerical value a point Drawbacks 3 22 11 How different is the estimate from the true parameter How reliable is your estimate How confident are you with your estimate Ways to improve Lecture 16 4 Confidence Interval A confidence interval has the form point estimate margin of error The point estimate is our guess for the value of an unknown parameter The margin of error shows how accurate we believe our guess is based on the sampling distribution of the estimate C confidence level which shows how confident we are for the confidence interval to cover the true population mean 3 22 11 Lecture 16 5 Confidence Interval for m We are interested in estimating the population mean m The population SD is assumed to be known To estimate m a sample of size n is drawn from the population and its mean x is calculated We know that x has approximately a normal distribution and x m Z N 0 1 n 3 22 11 Lecture 16 6 3 22 11 Lecture 16 7 Then x m P z z C n P m z x m z C n n This leads to P x z m x z C n n Thus a level C confidence interval for m is x z x z n n 3 22 11 Lecture 16 8 TV Watching Time continued The number and the types of television programs and commercials targeted at children are affected by the amount of time children watch TV A survey was conducted among 100 children in which they were asked to record the number of hours they watched TV per week The sample mean is x 27 191 The population standard deviation of TV watch was known to be 8 0 Estimate the average watching time at 95 confidence level 3 22 11 Lecture 16 9 TV Watching Time continued The parameter to be estimated is m the average time that a child watches TV per week We need to compute the 95 confidence interval for m x z 27 191 1 96 3 22 11 n 27 191 z 8 0 100 8 0 27 191 1 57 25 621 28 761 100 Lecture 16 10 Inventory Cost To lower inventory costs a computer company wants to employ an inventory model Demand during lead time is normally distributed with a s d of 50 computers It is required to know the mean in order to calculate optimum inventory levels Demand during 60 lead times has x 499 75 Estimate the mean demand during lead time with 95 confidence The 95 confidence interval is x z 3 22 11 50 499 75 1 96 499 75 12 65 487 1 512 4 n 60 Lecture 16 11 How should we understand and interpret CI A 95 confidence interval CI means that the confidence interval is calculated by a method that will cover the true value in 95 of all possible samples For a given sample whether the CI covers the true value is known i e no uncertainty Imagine there are 100 repeated samples Based on each sample a 95 CI can be constructed There will be approximately 95 CI s that will cover the true mean A common wrong statement The CI will cover the true value with probability 95 3 22 11 Lecture 16 12 3 22 11 Lecture 16 13 Four commonly used confidence levels Confidence level 0 90 0 95 0 98 0 99 3 22 11 Z 1 645 1 96 2 33 2 575 Lecture 16 14 Margin of Error The length of a CI is given by 2z n The margin of error is half of the length z n Margin of error is a measure of precision or accuracy The smaller the more accurate 3 22 11 Lecture 16 15 Precision The margin of error is a function of the population standard deviation the confidence level the sample size z n If everything else remains the same then The larger the sample size the narrower the CI The higher the confidence level the wider the CI The larger the population SD the wider the CI 3 22 11 Lecture 16 16 Selecting the Sample Size A common strategy is to first specify both desired confidence level reliability margin of error accuracy Then determine the necessary sample size n as follows The phrase estimate the mean to within W units translates to a confidence interval of the form x W The required sample size to estimate the mean is 3 22 11 z n W 2 Lecture 16 17 Lumber Production To estimate the amount of lumber that can be harvested in an area the mean diameter of trees in the area must be estimated to within one inch with 99 confidence What sample size should be taken Assume diameters are normally distributed with 6 inches The margin of error is 1 inch i e W 1 The confidence level 99 leads to the z score 2 575 z n W 3 22 11 2 2 575 6 239 1 2 Lecture 16 18 Response Time Suppose that the response time to a particular editing command is normally distributed with standard deviation 25 milliseconds What sample size is necessary to ensure that the 95 CI for m has margin of error of at most 5 Note W 5 The sample size n satisfies 5 1 96 25 n Solving for n we have n 1 96 25 5 9 80 96 04 2 2 Since n must be an integer a sample size of 97 is required 3 22 11 Lecture 16 19 Take Home Message Point estimation Confidence interval Definition Interpretation Confidence interval for a population mean Margin of error and sample size determination 3 22 11 Lecture 16 20
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