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Chapter 9: Problem 41

Example \(9.3\) gave the following airborne times for United Airlines flight 448 from Albuquerque to Denver on 10 randomly selected days: \(\begin{array}{llllllllll}57 & 54 & 55 & 51 & 56 & 48 & 52 & 51 & 59 & 59\end{array}\) a. Compute and interpret a \(90 \%\) confidence interval for the mean airborne time for flight 448 . b. Give an interpretation of the \(90 \%\) confidence level associated with the interval estimate in Part (a). c. Based on your interval in Part (a), if flight 448 is scheduled to depart at 10 A.M., what would you recommend for the published arrival time? Explain.

Short Answer

Expert verified
Based on the 90% confidence interval computed for the mean airborne time, it's recommended that the published arrival time should take into account the upper limit of this interval to allow for any possible deviations in flight times.

Step by step solution

01

Calculate the Mean

The first step is to calculate the mean (average) of the given airborne times. To do this, we add up all the times and divide by the total number, which is 10. This will give us our mean airborne time.
02

Compute the Standard Deviation

The next step is to compute the standard deviation, which gives us a measure of how spread out the numbers are. We subtract the mean from each number, square the result, sum up these squares, divide by the count minus one (to get the variance), and finally take the square root to obtain the standard deviation.
03

Calculate the Standard Error

Next, we need to calculate the standard error, which is the standard deviation divided by the square root of the sample size.
04

Calculate the Confidence Interval

Now calculate the 90% confidence interval. For this, multiply the standard error by the value of the t statistic for the desired confidence level (90% in this case). Subtract this value from the mean for the lower limit and add it for the upper limit. This will give us the 90% confidence interval.
05

Interpret the Confidence Interval

Now interpret the interval. A 90% confidence interval means that we are 90% confident that the true population mean falls within this range.
06

Make a Recommendation for the Arrival Time

Based on our interval, we should consider the upper limit for the arrival time to accommodate any possible variations in flight times, thus ensuring that the plane arrives within the promised timeframe most of the time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Analysis
Statistical analysis involves collecting, examining, and interpreting data to discover patterns and relationships. It is a key component in making informed decisions based on numerical data. For instance, in the provided exercise, statistical analysis allows us to estimate flight time durations for United Airlines flight 448 using a sample of data. By applying statistical concepts such as mean, standard deviation, and confidence intervals, we can predict and plan for future flight schedules. Such predictive analysis is vital for optimizing operational efficiency and improving customer satisfaction in various industries.
Mean Airborne Time
The mean airborne time is essentially the average time a flight spends in the air. In the context of the exercise, the mean is calculated by adding the airborne time for the sampled flights and dividing by the number of flights. This average represents a central value for our data set and serves as a point of reference. It is important to note that while the mean is informative, it does not tell us about the variability of the flight times which is where the standard deviation and standard error come into play.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. In simpler terms, it tells us how much the airborne times deviate, on average, from the mean airborne time. A low standard deviation indicates that the times are close to the mean, whereas a high standard deviation indicates that the times are spread out over a larger range of values. In real-world applications, understanding standard deviation helps in predicting the reliability and consistency of certain processes or events.
Standard Error
The standard error is an indication of how much the sample mean of the data is likely to differ from the true population mean. It's calculated using the standard deviation and the sample size. The smaller the standard error, the more precise the estimate of the population mean. In our exercise, by calculating the standard error, we are able to create a confidence interval which provides a range that is likely to contain the true mean time for all flights 448. This helps in determining schedule times that are more likely to be accurate, reducing the risk of delays and improving scheduling strategies.

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Most popular questions from this chapter

In 1991, California imposed a "snack tax" (a sales \(\operatorname{tax}\) on snack food) in an attempt to help balance the state budget. A proposed alternative tax was a \(12 \phi\) -per-pack increase in the cigarette tax. In a poll of 602 randomly selected California registered voters, 445 responded that they would have preferred the cigarette tax increase to the snack tax (Reno Gazette-Journal, August 26,1991 ). Estimate the true proportion of California registered voters who preferred the cigarette tax increase; use a \(95 \%\) confidence interval.

Data consistent with summary quantities in the article referenced in Exercise \(9.3\) on total calorie consumption on a particular day are given for a sample of children who did not eat fast food on that day and for a sample of children who did eat fast food on that day. Assume that it is reasonable to regard these samples as representative of the population of children in the United States. No Fast Food \(\begin{array}{llllllll}2331 & 1918 & 1009 & 1730 & 1469 & 2053 & 2143 & 1981 \\ 1852 & 1777 & 1765 & 1827 & 1648 & 1506 & 2669 & \\ \text { Fast Food } & & & & & & \\ 2523 & 1758 & 934 & 2328 & 2434 & 2267 & 2526 & 1195 \\ 890 & 1511 & 875 & 2207 & 1811 & 1250 & 2117 & \end{array}\) a. Use the given information to estimate the mean calorie intake for children in the United States on a day when no fast food is consumed. b. Use the given information to estimate the mean calorie intake for children in the United States on a day when fast food is consumed. c. Use the given information to estimate the produce estimates of the standard deviations of calorie intake for days when no fast food is consumed and for days when fast food is consumed.

The eating habits of 12 bats were examined in the article "Foraging Behavior of the Indian False Vampire Bat" (Biotropica \([1991]: 63-67) .\) These bats consume insects and frogs. For these 12 bats, the mean time to consume a frog was \(\bar{x}=21.9 \mathrm{~min}\). Suppose that the standard deviation was \(s=7.7 \mathrm{~min} .\) Construct and interpret a \(90 \%\) confidence interval for the mean suppertime of a vampire bat whose meal consists of a frog. What assumptions must be reasonable for the one-sample \(t\) interval to be appropriate?

The following data on gross efficiency (ratio of work accomplished per minute to calorie expenditure per minute) for trained endurance cyclists were given in the article "Cycling Efficiency Is Related to the Percentage of Type I Muscle Fibers" (Medicine and Science in Sports and Exercise [1992]: \(782-88\) ): \(\begin{array}{llllllll}18.3 & 18.9 & 19.0 & 20.9 & 21.4 & 20.5 & 20.1 & 20.1\end{array}\) \(\begin{array}{llllllll}20.8 & 20.5 & 19.9 & 20.5 & 20.6 & 22.1 & 21.9 & 21.2\end{array}\) \(\begin{array}{lll}20.5 & 22.6 & 22.6\end{array}\) a. Assuming that the distribution of gross energy in the population of all endurance cyclists is normal, give a point estimate of \(\mu\), the population mean gross efficiency. b. Making no assumptions about the shape of the population distribution, estimate the proportion of all such cyclists whose gross efficiency is at most 20 .

Because of safety considerations, in May 2003 the Federal Aviation Administration (FAA) changed its guidelines for how small commuter airlines must estimate passenger weights. Under the old rule, airlines used \(180 \mathrm{lb}\) as a typical passenger weight (including carry-on luggage) in warm months and \(185 \mathrm{lb}\) as a typical weight in cold months. The Alaska Journal of Commerce (May 25, 2003 ) reported that Frontier Airlines conducted a study to estimate average passenger plus carry-on weights. They found an average summer weight of \(183 \mathrm{lb}\) and a winter average of \(190 \mathrm{lb} .\) Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 20 lb for the summer weights and \(23 \mathrm{lb}\) for the winter weights. a. Construct and interpret a \(95 \%\) confidence interval for the mean summer weight (including carry-on luggage) of Frontier Airlines passengers. b. Construct and interpret a \(95 \%\) confidence interval for the mean winter weight (including carry-on luggage) of Frontier Airlines passengers. c. The new FAA recommendations are \(190 \mathrm{lb}\) for summer and \(195 \mathrm{lb}\) for winter. Comment on these recommendations in light of the confidence interval estimates from Parts (a) and (b).
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