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Chapter 9: Problem 48

Example \(9.3\) gave the following airborne times (in minutes) for 10 randomly selected flights from San Francisco to Washington Dulles airport: \(\begin{array}{llllllllll}270 & 256 & 267 & 285 & 274 & 275 & 266 & 258 & 271 & 281\end{array}\) a. Compute and interpret a \(90 \%\) confidence interval for the mean airborne time for flights from San Francisco to Washington Dulles. b. Give an interpretation of the \(90 \%\) confidence level associated with the interval estimate in Part (a). c. If a flight from San Francisco to Washington Dulles is scheduled to depart at 10 A.M., what would you recommend for the published arrival time? Explain.

Short Answer

Expert verified
The computed 90% confidence interval is an estimate of the range within which the true average flight time is likely to fall. For a 10 A.M. departure, it's best to plan for an arrival time based on the upper limit of this interval.

Step by step solution

01

Computing Mean Airborne Time

Compute the mean or average of the given flight times. The mean is computed by adding up all the numbers and then dividing by the number of numbers. In this case, add all ten flight times and then divide by 10.
02

Computing Standard Deviation and Standard Error

Compute the standard deviation, which provides information about the dispersion or spread of the flight times. To compute standard deviation, subtract the mean from each individual flight time (resulting in deviation), square these deviations to make them positive, add all the squared deviations together, and then take the square root. To find the standard error, divide by the square root of the number of flights.
03

Computing 90% Confidence Interval

Use the standard error and a z-value from a standard normal distribution corresponding to your desired level of confidence (90% in this case, which corresponds to a z-value of 1.645) to calculate the confidence interval. Subtract the result from the mean for the lower bound, and add the result to the mean for the upper bound.
04

Interpreting the Confidence Interval

This interval provides an estimation of the range in which we are 90% confident the true average flight time lies.
05

Recommendation for Arrival Time

For the flight's scheduled departure at 10 A.M., it would be safest to advise an arrival time based on the upper limit of the confidence interval computed in step 3. This conservative estimate accommodates for variability and potential delays.

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Key Concepts

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

Mean Airborne Time
Mean airborne time is an essential metric for airlines and travelers, representing the average time a plane spends in the air between two destinations. It is calculated simply by adding the total amount of time spent in the air for a group of flights and then dividing by the number of flights. For example, if we have time records of ten flights, we sum all these times and divide by 10 to find the mean.

In practical terms, understanding the mean airborne time helps airlines to schedule flights and manage timetables efficiently. It's also useful for passengers planning their travel, as it gives an idea of how long they'll be in the air.
Standard Deviation
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion from the average (mean). In the context of flight times, the standard deviation will show us how much the airborne time for flights varies from the mean airborne time.

A small standard deviation indicates that the flight times are closely clustered around the mean, while a large standard deviation suggests a wide range of airborne times. For airlines and travelers, a lower standard deviation implies predictability in flight duration, while a higher value may require planning for more variability.
Standard Error
The standard error of the mean (SEM) further refines our understanding of data spread by describing how far the sample mean of the data is likely to be from the true population mean. The standard error is calculated by dividing the standard deviation by the square root of the sample size.

As the sample size in any data set increases, the standard error decreases, suggesting a more accurate estimate of the population mean. A low standard error in flight times indicates that the sample mean is a reliable estimate of the population mean.
Statistical Inference
Statistical inference allows us to draw conclusions about a population based on a sample from that population. Using the concepts of mean airborne time, standard deviation, and standard error, we can perform statistical inference through the computation of confidence intervals.

A confidence interval in this context gives a range within which we expect the true mean airborne time for all flights from San Francisco to Washington Dulles to fall, with a certain degree of certainty, say 90%. This interval estimates the true mean and informs both airlines and travelers about expected flight duration with some assuredness.
Data Analysis
Data analysis involves inspecting, cleaning, transforming, and modeling data with the goal of discovering useful information. When analyzing flight times, we first calculate basic descriptive statistics like the mean, standard deviation, and standard error. Then, we may create confidence intervals to inform our understanding of the data.

In the scenario provided, data analysis not only calculates a confidence interval but also interprets what that interval means in practical terms for scheduling a flight鈥檚 arrival time. This kind of analysis is critical in making informed decisions and recommendations based on statistical data.

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Most popular questions from this chapter

Why is an unbiased statistic generally preferred over a biased statistic for estimating a population characteristic? Does unbiasedness alone guarantee that the estimate will be close to the true value? Explain. Under what circumstances might you choose a biased statistic over an unbiased statistic if two statistics are available for estimating a population characteristic?

An Associated Press article on potential violent behavior reported the results of a survey of 750 workers who were employed full time (San Luis Obispo Tribune, September 7. 1999). Of those surveyed, 125 indicated that they were so angered by a coworker during the past year that they felt like hitting the coworker (but didn't). Assuming that it is reasonable to regard this sample of 750 as a random sample from the population of full-time workers, use this information to construct and interpret a \(90 \%\) confidence interval estimate of \(p\), the true proportion of full-time workers so angered in the last year that they wanted to hit a colleague.

The article "Hospitals Dispute Medtronic Data on Wires" (The Wall Street journal. February 4,2010 ) describes several studies of the failure rate of defibrillators used in the treatment of heart problems. In one study conducted by the Mayo Clinic, it was reported that failures were experienced within the first 2 years by 18 of 89 patients under 50 years old and 13 of 362 patients age 50 and older who received a particular type of defibrillator. Assume it is reasonable to regard these two samples as representative of patients in the two age groups who receive this type of defibrillator. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator. b. Construct and interpret a \(99 \%\) confidence interval for the proportion of patients age 50 and older who experience a failure within the first 2 years after receiving this type of defibrillator. c. Suppose that the researchers wanted to estimate the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator to within \(.03\) with \(95 \%\) confidence. How large a sample should be used? Use the results of the study as a preliminary estimate of the population proportion.

The article "The Association Between Television Viewing and Irregular Sleep Schedules Among Children Less Than 3 Years of Age" (Pediatrics [2005]: \(851-856\) ) reported the accompanying \(95 \%\) confidence intervals for average TV viewing time (in hours per day) for three different age groups. $$\begin{array}{lc} \text { Age Group } & 95 \% \text { Confidence Interval } \\ \hline \text { Less than } 12 \text { months } & (0.8,1.0) \\ 12 \text { to } 23 \text { months } & (1.4,1.8) \\ 24 \text { to } 35 \text { months } & (2.1,2.5) \\ \hline \end{array}$$ a. Suppose that the sample sizes for each of the three age group samples were equal. Based on the given confidence intervals, which of the age group samples had the greatest variability in TV viewing time? Explain your choice. b. Now suppose that the sample standard deviations for the three age group samples were equal, but that the three sample sizes might have been different. Which of the three age-group samples had the largest sample size? Explain your choice. c. The interval \((.768,1.032)\) is either a \(90 \%\) confidence interval or a \(99 \%\) confidence interval for the mean TV viewing time computed using the sample data for children less than 12 months old. Is the confidence level for this interval \(90 \%\) or \(99 \%\) ? Explain your choice.

The authors of the paper "Deception and Design: The Impact of Communication Technology on Lying Behavior" (Proceedings of Computer Human Interaction [2004]) asked 30 students in an upper division communications course at a large university to keep a journal for 7 days, recording each social interaction and whether or not they told any lies during that interaction. A lie was defined as "any time you intentionally try to mislead someone." The paper reported that the mean number of lies per day for the 30 students was \(1.58\) and the standard deviation of number of lies per day was \(1.02 .\) a. What assumption must be made in order for the \(t\) confidence interval of this section to be an appropriate method for estimating \(\mu\), the mean number of lies per day for all students at this university? b. Would you recommend using the \(t\) confidence interval to construct an estimate of \(\mu\) as defined in Part (a)? Explain why or why not.
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