Chapter 15 Confidence Intervals Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters Before deciding to offer an affinity credit card to alumni of a university, the credit company wants to know how many customers will accept the offer and how large a balance they will carry? Use confidence intervals to answer such questions They convey information about the precision of the estimates of 41 Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters Two Parameters of Interest p, the proportion who will return the application for the credit card µ, the average monthly balance that those who accept the credit card will carry of 41 Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters Summary Statistics (n = 1000) of 41 Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters Confidence Interval for the Proportion A confidence interval is a range of plausible values for a parameter based on a sample Constructing confidence intervals relies on the sampling distribution of the statistic of 41 Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters Confidence Interval for the Proportion The Central Limit Theorem implies a normal model for the sampling distribution of ˆ p E( ) = p and SE( ˆ p )= ˆ p p (1 − p ) / n of 41 Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters 95% Confidence Interval for p The sample statistic in 95% of samples lies within 1.96 standard errors of the population parameter of 41 Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters 95% Confidence Interval for p For any of these samples, the interval formed by reaching 1.96 standard errors to the left and right of will contain p p ˆ The estimated standard error (se) is used in constructing the confidence interval (i.e., is substituted for p) ˆ p of 41 Copyright © 2011 Pearson Education, Inc 15.1 Ranges for Parameters 95% Confidence Interval for p The 100(1 – α)% confidence interval for p is pˆ − za /2 pˆ ( − pˆ ) / n to pˆ + za /2 pˆ ( − pˆ ) / n For a 95% confidence interval zα/2 = 1.96 10 of 41 Copyright © 2011 Pearson Education, Inc 15.5 Margin of Error A precise confidence interval has a small margin of error Margin of error is affected by (1) level of confidence, (2) variation in the data and (3) number of observations It is used in determining sample size 27 of 41 Copyright © 2011 Pearson Education, Inc 15.5 Margin of Error Determining Sample Size For a study about µ with 95% coverage, find the sample size using n = 4σ2 / (Margin of Error)2 Obtain an estimate for σ2 using a pilot sample (since we have to choose n before collecting data) 28 of 41 Copyright © 2011 Pearson Education, Inc 15.5 Margin of Error Example A nutritionist wants to know the average calorie intake for female customers to within ± 50 calories with 95% confidence A pilot study gives an estimate of 430 calories for σ Find n n = 4(4302) / 502 = 295.8 or 300 customers 29 of 41 Copyright © 2011 Pearson Education, Inc 15.5 Margin of Error Determining Sample Size For a study about p, no need for a pilot sample Use p = 0.5 which results in largest possible value for σ of ½ 30 of 41 Copyright © 2011 Pearson Education, Inc 15.5 Margin of Error Sample Sizes for Various Margins of Error (95% coverage) 31 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.1: PROPERTY TAXES Motivation A mayor is considering a tax on business that is proportional to the amount spent to lease property in his city How much revenue would a 1% tax generate? 32 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.1: PROPERTY TAXES Method Need a confidence interval for µ (average cost of a lease) to obtain a confidence interval for the amount raised by the tax Check conditions (SRS and sample size) before proceeding 33 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.1: PROPERTY TAXES Mechanics: Statistics on Lease Costs 34 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.1: PROPERTY TAXES Message We are 95% confident that the average cost of a lease is between $410,000 and $550,000 The 95% confidence interval for tax raised per business is therefore [$4,100 to $5,500] Since the number of businesses leased in the city is 4,500, we are 95% confident that the amount raised will be $18,450,000 to $24,750,000 35 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.2: A POLITICAL POLL Motivation The mayor is seeking reelection Only 40% of registered voters think he is doing a good job (n = 400) What does this indicate about the attitudes of all voters in the city? 36 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.2: A POLITICAL POLL Method Construct a 95% confidence interval for the population proportion, p Check SRS and sample size conditions 37 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.2: A POLITICAL POLL Mechanics Find the estimated standard error, se ( ) se ( ) = = 0.0245 ˆ p (0.4)(0.6) 400 The 95% p confidence interval is ˆ 0.40 ± 1.96 (0.0245) = [0.352 to 0.448] 38 of 41 Copyright © 2011 Pearson Education, Inc 4M Example 15.2: A POLITICAL POLL Message The mayor can be 95% certain that 35% to 45% of registered voters think he is doing a good job 39 of 41 Copyright © 2011 Pearson Education, Inc Best Practices Be sure that the data are an SRS from the population Stick to 95% confidence intervals Round the endpoints of intervals when presenting the results Use full precision for intermediate calculations 40 of 41 Copyright © 2011 Pearson Education, Inc Pitfalls Do not claim that a 95% confidence interval holds µ Do not use a confidence interval to describe other samples Do not manipulate the sampling to obtain a particular confidence interval 41 of 41 Copyright © 2011 Pearson Education, Inc ... Pearson Education, Inc 15. 1 Ranges for Parameters 95% Confidence Interval for p For any of these samples, the interval formed by reaching 1.96 standard errors to the left and right of will contain... Pearson Education, Inc 15. 1 Ranges for Parameters Summary Statistics (n = 1000) of 41 Copyright © 2011 Pearson Education, Inc 15. 1 Ranges for Parameters Confidence Interval for the Proportion ... Education, Inc 15. 1 Ranges for Parameters Checklist for Confidence Interval for p SRS condition The sample is a simple random sample from the relevant population Sample size condition (for proportion)