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Although airline schedules and cost are important factors for business travelers whenchoosing an airline carrier, a USA Today survey found that business travelers list anairline's frequent flyer program as the most important factor. From a sample of $n=1993$ business travelers who responded to the survey, 618 listed a frequent flyer program as themost important factor.$$\begin{array}{l}{\text { a. What is the point estimate of the proportion of the population of business travelers }} \\ {\text { who believe a frequent flyer program is the most important factor when choosing an }} \\ {\text { airline carrier? }}\end{array}$$$$\begin{array}{l}{\text { b. Develop a } 95 \% \text { confidence interval estimate of the population proportion. }} \\ {\text { c. How large a sample would be required to report the margin of error of } 01 \text { at } 95 \%} \\ {\text { confidence? Would you recommend that USA Today attempt to provide this degree of }} \\ {\text { precision? Why or why not? }}\end{array}$$

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a. 0.3101b. 0.2898 to 0.3304c. 8219

Intro Stats / AP Statistics

Chapter 8

Interval Estimation

Confidence Intervals

Kaniz J.

April 15, 2022

Although airline schedules and cost are important factors for business travelers when choosing an airline carrier, a USA Today survey found that business travelers list an air- line’s frequent flyer program as the most important factor. From a sample of

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all right. This question deals with a survey in which 618 people said that frequent flyer benefits are the most important factor to them in choosing an airline. So part eh wants us to find the point estimate for the proportion of people that we'll say that frequent flyer is the most important reason for choosing an airline. So p hat. It's simply the number of successes divided by the sample size, which in this case 6 18 said the frequent flyer benefits were the most important factor divided by our total number of people samples. And this works out to be point 3101 Then Part B wants us to calculate the 95% confidence interval. So for a proportion, the confidence interval is the point estimate, plus or minus our special Z score. Times Square root of P Hat Times Q hat all over end, which in this case, since we're dealing with 95% confidence, we know all the variables. 0.3101 for a P hat, plus or minus. Our special Z score in this case is 1.96 times square root of he had times one minus P hat all over the sample size. And this gives us confidence interval ranging from point to a 98 up to 0.3304 So we're 95% confident that the true proportion of flyers that say that frequent flyer benefits are the most important factor are between these two values. Then part see says, we want a margin of error of 0.1 s. So what's the sample size we need still sticking with 95% confidence? So we use our equation. Sample size proportions equals P Hat times Q Hat times Z star over margin of error squared, which in this case we know RPS is 0.3101 times air Q had, which is one minus that times ours e star divided by our margin of error squared. And this works out to be 82 18 0.6, which we always round up on these questions to ensure that our margin of error is met and it asks if we recommend the airline need this amount of precision and if you look 8000 people is a lot to survey and are 95 confidence interval for the sample. Already it's basically between 0.29 and 0.33 So I'd say this degree of precision is not necessary. Is a likely not necessary because if you look at this proportion right here in our in precise confidence interval 0.292 point 33 that's already a semi narrow range and any marketing they'd use to target that range. I personally don't think it would make that big of a difference if they knew the exact proportion better because I feel like if they were going to make any marketing decision based off of a confidence interval, it's not going to make a big difference. If it's 0.295 versus 0.3 to 5. I think that's a narrow enough arrange that they have an idea of what marketing strategy to use

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