2 A company representative claims that 45 percent of Burger King sales are made

2 a company representative claims that 45 percent of

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2. A company representative claims that 45 percent of Burger King sales are made at the drive-through window. 3. A survey of homes in the Chicago area indicated that 85 percent of the new construction had central air conditioning. 4. A recent survey of married men between the ages of 35 and 50 found that 63 percent felt that both partners should earn a living. 305
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Using the Normal Distribution to Approximate the Binomial Distribution To develop a confidence interval for a proportion, we need to meet the following assumptions. 1. The binomial conditions, discussed in Chapter 6, have been met. Briefly, these conditions are: a. The sample data is the result of counts. b. There are only two possible outcomes. c. The probability of a success remains the same from one trial to the next. d. The trials are independent. This means the outcome on one trial does not affect the outcome on another. 2. The values n ʌ and n (1- ʌ ) should both be greater than or equal to 5. This condition allows us to invoke the central limit theorem and employ the standard normal distribution, that is, z , to complete a confidence interval. 306 Assumptions when using a Normal approximation for confidence intervals with a proportion ( short version ) 1. It meets the conditions of a BINOMIAL distribution 2. Both n* ʌ and n*(1- ʌ ) are at least 5 306
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Confidence Interval for a Population Proportion - Formula 306 Confidence Interval for a Population Proportion - Example The union representing the Bottle Blowers of America (BBA) is considering a proposal to merge with the Teamsters Union. According to BBA union bylaws, at least three-fourths of the union membership must approve any merger. A random sample of 2,000 current BBA members reveals 1,600 plan to vote for the merger proposal. What is the estimate of the population proportion? Develop a 95 percent confidence interval for the population proportion. Basing your decision on this sample information, can you conclude that the necessary proportion of BBA members favor the merger? Why? . membership union the of percent 75 than greater values includes estimate interval the because pass likely will proposal merger The : Conclude ) 818 . 0 , 782 . 0 ( 018 . 80 . 2,000 ) 80 . 1 ( 80 . 96 . 1 80 . 0 ) 1 ( C.I. C.I. 95% the Compute 80 . 0 2000 1,600 : proportion sample the compute First, 2 / ! r ! 0 r ! 0 r ! ! ! ! n p p z p n x p D 307
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Finite-Population Correction Factor z A population that has a fixed upper bound is said to be finite. z For a finite population, where the total number of objects is N and the size of the sample is n , the following adjustment is made to the standard errors of the sample means and the proportion: z However, if n / N < .05 , the finite-population correction factor may be ignored. 1 0 0 ! N n N n x V V 1 ) 1 ( 0 0 0 ! N n N n p p p V Standard Error of the Mean Standard Error of the Proportion 309 Effects on FPC when n/N Changes Observe that FPC approaches 1 when n/N becomes smaller 309
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Confidence Interval Formulas for Estimating Means and Proportions with Finite Population Correction 1 0 0 r N n N n z X V C.I. for the Mean ( P ) 1 ) 1 ( 0 0 0 r N n N n p p z p C.I. for the Proportion ( S ) 1 0 0 r N n N n s t X C.I. for the Mean ( P ) 309
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