/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 In automobile mileage and gasoli... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 11

In automobile mileage and gasoline-consumption testing, 13 automobiles were road tested for 300 miles in both city and highway driving conditions. The following data were recorded for miles-per-gallon performance. \\[\text { City: } \quad 16.2 \quad 16.7 \quad 15.9 \quad 14.4 \quad 13.2 \quad 15.3 \quad 16.8 \quad 16.0 \quad 16.1 \quad 15.3 \quad 15.2 \quad 15.3 \quad 16.2\\] Highway: \(19.4 \quad 20.6 \quad 18.3 \quad 18.6 \quad 19.2 \quad 17.4 \quad 17.2 \quad 18.6 \quad 19.021 .1 \quad 19.4 \quad 18.5 \quad 18.7\) Use the mean, median, and mode to make a statement about the difference in performance for city and highway driving.

Short Answer

Expert verified
Highway driving shows better mileage than city driving based on mean, median, and mode.

Step by step solution

01

Calculate the Mean for City Driving

The mean is the average of all data points. Add up all the city MPG values and divide by the number of values, which is 13. \[\text{Mean}_{\text{City}} = \frac{16.2 + 16.7 + 15.9 + 14.4 + 13.2 + 15.3 + 16.8 + 16.0 + 16.1 + 15.3 + 15.2 + 15.3 + 16.2}{13} = \frac{192.4}{13} \approx 14.8\]
02

Calculate the Mean for Highway Driving

Add up all the highway MPG values and divide by the number of values, which is 13.\[\text{Mean}_{\text{Highway}} = \frac{19.4 + 20.6 + 18.3 + 18.6 + 19.2 + 17.4 + 17.2 + 18.6 + 19.0 + 21.1 + 19.4 + 18.5 + 18.7}{13} = \frac{245.0}{13} \approx 18.8\]
03

Determine the Median for City Driving

The median is the middle value when the data points are ordered. Arrange the city MPG values in ascending order and find the middle one.Ordered data: \[13.2, 14.4, 15.2, 15.3, 15.3, 15.3, 15.9, 16.0, 16.1, 16.2, 16.2, 16.7, 16.8\]With 13 values, the median is the 7th value: 15.9.
04

Determine the Median for Highway Driving

Order the highway MPG values and find the middle one.Ordered data: \[17.2, 17.4, 18.3, 18.5, 18.6, 18.6, 18.7, 19.0, 19.2, 19.4, 19.4, 20.6, 21.1\]With 13 values, the median is the 7th value: 18.7.
05

Identify the Mode for City Driving

The mode is the most frequently occurring value. In the city data, 15.3 appears three times, more often than any other value.
06

Identify the Mode for Highway Driving

Check the frequency of each value. In the highway data, 18.6 and 19.4 each appear twice, making both the modes. This dataset is bimodal.
07

Interpret the Results

The mean and median suggest that highway mileage is better than city mileage (mean: 14.8 city vs 18.8 highway, median: 15.9 city vs 18.7 highway). The mode supports this as well (city mode: 15.3, highway modes: 18.6 and 19.4), indicating that typical highway performance is superior.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Mean Calculation
The mean, often referred to as the average, is a fundamental concept in descriptive statistics used to summarize a set of numbers.
To calculate the mean, you need to add up all the figures and then divide by the number of values in the data set. This step gives you a single value that represents the data set.
For instance, in the exercise on automobile mileage, we calculate the mean for both city and highway driving to compare the performances.
  • City Driving Mean: The sum of all city miles-per-gallon (MPG) data points is 192.4. Dividing this by 13 cars gives a mean of approximately 14.8 MPG.
  • Highway Driving Mean: By summing highway MPG data points, we get 245.0, and dividing by 13 gives a mean of approximately 18.8 MPG.
Understanding the mean helps in detecting the overall trend. Here, it shows that generally, cars perform better in highway conditions than in city driving.
Median Calculation
The median is a measure that indicates the middle value of a dataset, providing a central tendency indicator.
This concept is especially useful in datasets with outliers, as it is not skewed by extremely high or low values.
To find the median, arrange the data points in ascending order, then locate the middle value. In a dataset with an odd number of values, the median is straightforward.
  • City Median: Once the city MPG data is ordered, the median is the 7th value, which is 15.9 MPG.
  • Highway Median: With the highway MPG data in order, the median falls at 18.7 MPG, which is the 7th value in this dataset.
The median confirms findings from the mean calculation, highlighting that typical performance is better on highways.
Mode Identification
The mode represents the value that appears most frequently in a dataset. It is an essential measure of central tendency.
Sometimes, a dataset can have no mode, one mode, or multiple modes, known as bimodal or multimodal data.
Identifying the mode helps in understanding the most common performance metric among measurements.
  • City Driving Mode: In the city MPG data, 15.3 appears three times, making it the mode.
  • Highway Driving Mode: For highway MPG, the values 18.6 and 19.4 each appear twice, hence the dataset is bimodal.
Having these modes allows us to see which figures are most common and to note that highway MPG data shows multiple typical values, reinforcing better overall performance compared to city driving.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Small business owners often look to payroll service companies to handle their employee payroll. Reasons are that small business owners face complicated tax regulations and penalties for employment tax errors are costly. According to the Internal Revenue Service, \(26 \%\) of all small business employment tax returns contained errors that resulted in a tax penalty to the owner (The Wall Street Journal, January 30,2006 ). The tax penalties for a sample of 20 small business owners follow: \\[ \begin{array}{llllllll} 820 & 270 & 450 & 1010 & 890 & 700 & 1350 & 350 & 300 & 1200 \\ 390 & 730 & 2040 & 230 & 640 & 350 & 420 & 270 & 370 & 620 \end{array}\\] a. What is the mean tax penalty for improperly filed employment tax returns? b. What is the standard deviation? c. Is the highest penalty, \(\$ 2040,\) an outlier? d. What are some of the advantages of a small business owner hiring a payroll service company to handle employee payroll services, including the employment tax returns?

Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. \\[\begin{array}{rrrrrr} 8408 & 1374 & 1872 & 8879 & 2459 & 11413 \\ 608 & 14138 & 6452 & 1850 & 2818 & 1356 \\ 10498 & 7478 & 4019 & 4341 & 739 & 2127 \\ 3653 & 5794 & 8305 & &\end{array}\\] a. Provide a five-number summary. b. Compute the lower and upper limits. c. Do the data contain any outliers? d. Johnson \& Johnson's sales are the largest on the list at \(\$ 14,138\) million. Suppose a data entry error (a transposition) had been made and the sales had been entered as \(\$ 41,138\) million. Would the method of detecting outliers in part (c) identify this problem and allow for correction of the data entry error? e. Show a box plot.

The national average for the verbal portion of the College Board's Scholastic Aptitude Test (SAT) is 507 (The World Almanac, 2006 ). The College Board periodically rescales the test scores such that the standard deviation is approximately \(100 .\) Answer the following questions using a bell-shaped distribution and the empirical rule for the verbal test scores. a. What percentage of students have an SAT verbal score greater than \(607 ?\) b. What percentage of students have an SAT verbal score greater than \(707 ?\) c. What percentage of students have an SAT verbal score between 407 and \(507 ?\) d. What percentage of students have an SAT verbal score between 307 and \(607 ?\)

The U.S. Census Bureau provides statistics on family life in the United States, including the age at the time of first marriage, current marital status, and size of household (http://www.census.gov, March 20, 2006). The following data show the age at the time of first marriage for a sample of men and a sample of women. \\[ \begin{array}{lcccccccc} \text { Men } & 26 & 23 & 28 & 25 & 27 & 30 & 26 & 35 & 28 \\ & 21 & 24 & 27 & 29 & 30 & 27 & 32 & 27 & 25 \\ \text { Women } & 20 & 28 & 23 & 30 & 24 & 29 & 26 & 25 & \\ & 22 & 22 & 25 & 23 & 27 & 26 & 19 & & \end{array} \\] a. Determine the median age at the time of first marriage for men and women. b. Compute the first and third quartiles for both men and women. c. Twenty-five years ago the median age at the time of first marriage was 25 for men and 22 for women. What insight does this information provide about the decision of when to marry among young people today?

A department of transportation's study on driving speed and mileage for midsize automobiles resulted in the following data. $$\begin{array}{l|llllllllll} \text { Driving Speed } & 30 & 50 & 40 & 55 & 30 & 25 & 60 & 25 & 50 & 55 \\ \hline \text { Mileage } & 28 & 25 & 25 & 23 & 30 & 32 & 21 & 35 & 26 & 25 \end{array}$$ Compute and interpret the sample correlation coefficient.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.