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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 3 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 4 of 41 Copyright © 2011 Pearson Education, Inc. 15.1 Ranges for Parameters Summary Statistics (n = 1000) 5 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. 6 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 p̂. E( p̂) = p and SE( p̂) = p(1 p) / n 7 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. 8 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 p̂will contain p. The estimated standard error (se) is used in constructing the confidence interval (i.e., p̂ is substituted for p). 9 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 1 pˆ / n to pˆ za/2 pˆ 1 pˆ / n For a 95% confidence interval zα/2 = 1.96. 10 of 41 Copyright © 2011 Pearson 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). Both np̂ and n(1 pˆ ) are larger than 10. 11 of 41 Copyright © 2011 Pearson Education, Inc. 15.1 Ranges for Parameters Credit Card Example The estimated standard error is se( p̂ ) = • 0.14(1 0.14) = 0.01097 1000 The 95% confidence interval is 0.14 ± 1.96(0.01097) = [0.1185 to 0.1615] 12 of 41 Copyright © 2011 Pearson Education, Inc. 15.1 Ranges for Parameters Credit Card Example With 95% confidence, the population proportion that will accept the offer is between about 12% and 16%. A larger sample size will reduce the estimated standard error resulting in a narrower interval (a more precise estimate of p). 13 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Confidence Interval for µ A similar procedure as that used for p is used to construct a confidence interval for µ. The estimated standard error for X is used. se( X ) = s / n 14 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Student’s t-Distribution Used because we substitute s for σ in the estimated standard error. A parameter called degrees of freedom (df = n-1) controls the shape of the distribution. As n increases, the t-distribution more closely resembles the standard normal distribution. 15 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Student’s t-Distribution Standard normal (black) and Student’s t-distributions (red) 16 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Confidence Interval for µ The 100(1 – α)% confidence interval for µ is x - tα/2, n-1 s / n to x + tα/2, n-1 s / n . The value of t depends on the level of confidence and n – 1 degrees of freedom. 17 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Checklist for Confidence Interval for µ SRS condition. The sample is a simple random sample from the relevant population. Sample size condition. The sample size is larger than 10 times the squared skewness and 10 times the absolute value of the kurtosis. 18 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Percentiles of the t-Distribution The t value for 95% confidence and 139 df = 1.98. 19 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Credit Card Example The estimated standard error is se ( X ) = $2,833.33 / 140 = $239.46 The 95% confidence interval is $1,990.50 ± 1.98($239.46) [$1,516.37 to $2,464.63] 20 of 41 Copyright © 2011 Pearson Education, Inc. 15.2 Confidence Interval for the Mean Credit Card Example • We are 95% confident that µ lies between $1,516.37 and $2,464.63. • Might µ be $1,250? It could be, but based on the sample results it’s not likely. 21 of 41 Copyright © 2011 Pearson Education, Inc. 15.3 Interpreting Confidence Intervals Common Confusions: Wrong Interpretations 95% of all customers keep a balance of $1,520 to $2,460. The mean balance of 95% of samples of 140 accounts will fall between $1,520 and $2,460. The mean balance is between $1,520 and $2,460. 22 of 41 Copyright © 2011 Pearson Education, Inc. 15.4 Manipulating Confidence Intervals Obtaining Ranges for Related Quantities If [L to U] is a 100(1 – α)% confidence interval for µ, then [c x L to c x U] is a 100 (1 – α)% confidence interval for c x µ and [c + L to c + U] is a 100(1 – α)% confidence interval for c + µ. 23 of 41 Copyright © 2011 Pearson Education, Inc. 15.4 Manipulating Confidence Intervals Changing the Problem Creating a new variable is preferable to combining confidence intervals. Consider the following: Let Y = profit earned from each customer. A customer who does not accept the card costs the bank $8. Each customer who accepts the card costs the bank $58 but the bank earns 10% on the revolving credit card balance. 24 of 41 Copyright © 2011 Pearson Education, Inc. 15.4 Manipulating Confidence Intervals Creating a New Variable Therefore, profit ($) earned from a customer is yi = -8 if offer is not accepted 0.10 (Balance) – 58 if offer is accepted. 25 of 41 Copyright © 2011 Pearson Education, Inc. 15.4 Manipulating Confidence Intervals Summary Statistics for Profit Earned For 100,000 offers, the 95% confidence interval for total profit is from $557,000 to $2,017,000. 26 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 ( p̂ ) se ( p̂ ) = (0.4)(0.6) 400 = 0.0245 The 95% confidence interval is 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.