Statistics for business decision making and analysis robert stine and foster chapter 15

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)