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Chapter 3: Problem 16

The following data represent the flight time (in minutes) of a random sample of seven flights from Las Vegas, Nevada, to Newark, New Jersey, on United Airlines. Compute the mean, median, and mode flight time. $$ 282,270,260,266,257,260,267 $$

Short Answer

Expert verified
Mean = 266.0, Median = 266, Mode = 260.

Step by step solution

01

- Arrange the Data in Ascending Order

First, sort the given flight times in ascending order: 257, 260, 260, 266, 267, 270, 282.
02

- Calculate the Mean

Calculate the mean by adding up all the flight times and then dividing by the number of flights.Mean = \( \frac{257 + 260 + 260 + 266 + 267 + 270 + 282}{7} \ = \frac{1862}{7} \ = 266.0 \)
03

- Determine the Median

The median is the middle value of the ordered data set. Since there are 7 values, the median is the 4th value.Median = 266.
04

- Identify the Mode

The mode is the value that appears most frequently in the data set. Here, the flight time 260 appears twice.Mode = 260.

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Key Concepts

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

Mean Calculation
The mean is one of the most common ways to represent the average of a data set. To find the mean of our flight times, we start by adding up all the times. When you add 257, 260, 260, 266, 267, 270, and 282, you get a total of 1862.

Next, you need to divide this total by the number of values there are in the data set. In our case, there are 7 flight times. Therefore, the mean is calculated as follows:

\[\text{Mean} = \frac{1862}{7} = 266.0 \]

The mean flight time is 266.0 minutes. Remember, the mean is helpful because it gives us a single number that summarizes all the values in our data set.
Median Calculation
The median is the middle value in an ordered data set. It represents the point at which half the data lies below and half lies above. First, we need to arrange our flight times in ascending order, which we already have as:

257, 260, 260, 266, 267, 270, 282.

Since there are 7 values in our data set, the median is the fourth value because it sits right in the middle.
\[\text{Median} = 266 \]

So, the median flight time is 266 minutes. The median is a useful measure of central tendency, especially in skewed distributions, as it is not affected by extremely high or low values.
Mode Calculation
The mode is the value that appears most frequently in a data set. In the context of our flight times, we need to find the number that occurs most often. Reviewing our sorted data set: 257, 260, 260, 266, 267, 270, 282, we observe that 260 appears twice while all other numbers appear only once.
\[\text{Mode} = 260 \]

Therefore, the mode of our data set is 260 minutes. The mode is helpful as it indicates the most common value in the data set, which can be particularly useful in understanding trends or patterns.

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Most popular questions from this chapter

According to the U.S. Census Bureau, the mean of the commute time to work for a resident of Boston, Massachusetts, is 27.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes to answer the following: (a) What minimum percentage of commuters in Boston has a commute time within 2 standard deviations of the mean? (b) What minimum percentage of commuters in Boston has a commute time within 1.5 standard deviations of the mean? What are the commute times within 1.5 standard deviations of the mean? (c) What is the minimum percentage of commuters who have commute times between 3 minutes and 51.6 minutes?

The following data represent the percentage of workers who carpool to work for the 50 states plus Washington, D.C. Note: The minimum observation of \(7.2 \%\) corresponds to Maine and the maximum observation of \(16.4 \%\) corresponds to Hawaii. $$ \begin{array}{rrrrrrrrr} \hline 7.2 & 8.5 & 9.0 & 9.4 & 10.0 & 10.3 & 11.2 & 11.5 & 13.8 \\ \hline 7.8 & 8.6 & 9.1 & 9.6 & 10.0 & 10.3 & 11.2 & 11.5 & 14.4 \\ \hline 7.8 & 8.6 & 9.2 & 9.7 & 10.0 & 10.3 & 11.2 & 11.7 & 16.4 \\ \hline 7.9 & 8.6 & 9.2 & 9.7 & 10.1 & 10.7 & 11.3 & 12.4 & \\ \hline 8.1 & 8.7 & 9.2 & 9.9 & 10.2 & 10.7 & 11.3 & 12.5 & \\ \hline 8.3 & 8.8 & 9.4 & 9.9 & 10.3 & 10.9 & 11.3 & 13.6 & \\ \hline \end{array} $$ (a) Find the five-number summary. (b) Construct a boxplot. (c) Comment on the shape of the distribution.

A histogram of a set of data indicates that the distribution of the data is skewed right. Which measure of central tendency will likely be larger, the mean or the median? Why?

Household Winter Temperature Often, frequency distributions are reported using unequal class widths because the frequencies of some groups would otherwise be small or very large. Consider the following data, which represent the daytime household temperature the thermostat is set to when someone is home for a random sample of 750 households. Determine the class midpoint, if necessary, for each class and approximate the mean and standard deviation temperature.

What makes the range less desirable than the standard deviation as a measure of dispersion?
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