6/3/10 Lecture 16 1STOR 155 Introductory StatisticsChapter 6: Introduction to InferenceLecture 16: Estimation with ConfidenceThe UNIVERSITY of NORTH CAROLINAat CHAPEL HILL6/3/10 Lecture 16 2• 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 testingIntroductionStatistical InferenceSample Population6/3/10 Lecture 16 3• 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 isTV Watching Time.191.27x6/3/10 Lecture 16 4• Use sample mean to estimate population mean.• A point estimator makes inference about a population by estimating the value of an unknownparameter using a single numerical value (a point).• Drawbacks:• How different is the estimate from the true parameter?• How reliable is your estimate?• How confident are you with your estimate?• Ways to improve?Point Estimation6/3/10 Lecture 16 5Confidence Interval• A confidence interval has the form:point estimate margin of error• A 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 samplingdistribution of the estimate.• C: confidence level, which shows how confident we are for the confidence interval to cover the true population mean.6/3/10 Lecture 16 6• 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 is calculated.• We know that has (approximately) a normal distribution, and 1). ,0(~ NnxZmxxConfidence Interval for m6/3/10 Lecture 16 76/3/10 Lecture 16 8• Then,.)(**CnzxnzP mm• This leads to.)(**CnzxnzxP m.)/(**CznxzP m• Thus, a level C confidence interval for m is],[**nzxnzx6/3/10 Lecture 16 9• 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• The population standard deviation of TV watch was known to be = 8.0.• Estimate the average watching time at 95% confidence level. TV Watching Time (continued).191.27x6/3/10 Lecture 16 10• 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.1000.8191.27**znzx TV Watching Time (continued) 761.28,621.2557.1191.271000.896.1191.27 6/3/10 Lecture 16 11• 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• Estimate the mean demand during lead time with 95% confidence. • The 95% confidence interval is:Inventory Cost.75.499x 4.512,1.48765.1275.499605096.175.499*nzx6/3/10 Lecture 16 12How 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%.6/3/10 Lecture 16 136/3/10 Lecture 16 14Four commonly used confidence levelsConfidence level0.90 1.6450.95 1.960.98 2.330.99 2.575Z*6/3/10 Lecture 16 15Margin of Error• The length of a CI is given by:• The margin of error is half of the length:• Margin of error is a measure of precision or accuracy. – The smaller, the more accurate.nz*2nz*6/3/10 Lecture 16 16Precision• The margin of error is a function of:– the population standard deviation– the confidence level– the sample size• 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. nz*6/3/10 Lecture 16 17• 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 formThe required sample size to estimate the mean isSelecting the Sample Size.Wx .2*Wzn6/3/10 Lecture 16 18• 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.Lumber Production.2391)6(575.222*Wzn6/3/10 Lecture 16 19Response 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 mhas margin of error of at most 5? • Note: W = 5• The sample size n satisfies• Solving for n, we have• Since n must be an integer, a sample size of 97 is required.5 1.96 25/ .n22[1.96 25/5] (9.80) 96.04.n 6/3/10 Lecture 16 20Take Home Message• Point estimation• Confidence interval– Definition– Interpretation– Confidence interval for a population mean• Margin of error and sample size
View Full Document
Unlocking...