Statistics Informed Decisions Using Data

Michael Sullivan III

Chapter 3 Numerically Summarizing Data - all with Video Answers

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Section 1

Measures of Central Tendency

00:53

Problem 1

What does it mean if a statistic is resistant?

Trinity Steen

Numerade Educator

01:54

In the 2009 Current Population Survey conducted by the U.S. Census Bureau, two average household incomes were reported: 49,777 dollar and 67,976 dollar. One of these averages is the mean and the other is the median. Which is the mean? Support your answer.

Christopher Stanley

Numerade Educator

00:56

Problem 3

The U.S. Department of Housing and Urban Development (HUD) uses the median to report the average price of a home in the United States. Why do you think HUD uses the median?

Trinity Steen

Numerade Educator

01:13

Problem 4

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?

Christopher Stanley

Numerade Educator

01:05

Problem 5

If a data set contains 10,000 values arranged in increasing order, where is the median located?

Narayan Hari

Numerade Educator

01:16

Problem 6

True or False: A data set will always have exactly one mode.

Narayan Hari

Numerade Educator

01:01

Problem 7

nd the population mean or sample mean as indicated.
$$\text { Sample: } 20,13,4,8,10$$

Narayan Hari

Numerade Educator

01:03

Problem 8

nd the population mean or sample mean as indicated.
$$\text { Sample: } 83,65,91,87,84$$

Sanchit Jain

Numerade Educator

01:03

Problem 9

nd the population mean or sample mean as indicated.
$$\text { Population: } 3,6,10,12,14$$

Narayan Hari

Numerade Educator

01:00

Problem 10

nd the population mean or sample mean as indicated.
$$\text { Population: } 1,19,25,15,12,16,28,13,6$$

Narayan Hari

Numerade Educator

01:02

Problem 11

For Super Bowl XLIV, CBS television sold 55 ad slots for a total revenue of roughly $165.5 million. What was the mean price per ad slot?

Narayan Hari

Numerade Educator

01:09

Problem 12

The median for the given set of six ordered data values is $26.5 .$ What is the missing value? 712.21 _______ 4150

Narayan Hari

Numerade Educator

01:51

Problem 13

Crash Test Results The Insurance Institute for Highway Safety crashed the 2010 Ford Fusion four times at 5 miles per hour. The costs of repair for each of the four crashes were $$\$ 2529, \$ 1889, \$ 2610, \$ 1073$$ Compute the mean, median, and mode cost of repair.

Narayan Hari

Numerade Educator

01:45

Problem 14

Cell Phone Use The following data represent the monthly cell phone bill for my wife's phone for six randomly selected months. $$\$ 35.34, \$ 42.09, \$ 39.43, \$ 38.93, \$ 43.39, \$ 49.26$$ Compute the mean, median, and mode phone bill.

Trinity Steen

Numerade Educator

01:37

Problem 15

15. Concrete Mix A certain type of concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The strength of concrete is measured by pouring the mix into casting cylinders 6 inches in diameter and 12 inches tall. The concrete is allowed to set for 28 days. The concrete's strength is then measured. The following data represent the strength of nine randomly selected casts (in psi). $$3960,4090,3200,3100,2940,3830,4090,4040,3780$$ Compute the mean, median, and mode strength of the concrete (in psi).

Trinity Steen

Numerade Educator

02:17

Problem 16

Flight Time 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 Continental Airlines. $$282,270,260,266,257,260,267$$ Compute the mean, median, and mode flight time.

Narayan Hari

Numerade Educator

01:34

Problem 17

For each of the three histograms shown, determine whether the mean is greater than, less than, or approximately equal to the median. Justify your answer. (FIGURE CAN'T COPY)

Trinity Steen

Numerade Educator

01:58

Problem 18

Match the histograms shown to the summary statistics: (FIGURE CAN'T COPY)

Trinity Steen

Numerade Educator

05:06

Problem 19

$\mathbf{p} \mathbf{H}$ in Water The acidity or alkalinity of a solution is measured using $\mathrm{pH} . \mathrm{A}$ pH less than 7 is acidic; a pH greater than - 19. $\mathbf{p} \mathbf{H}$ in Water The acidity or alkalinity of a solution is measured using $\mathrm{pH} . \mathrm{A}$ pH less than 7 is acidic; a pH greater than 7 is alkaline. The following data represent the $\mathrm{pH}$ in samples of bottled water and tap water.7 is alkaline. The following data represent the $\mathrm{pH}$ in samples of bottled water and tap water. (FIGURE CAN'T COPY)
(a) Determine the mean, median, and mode $p$ H for each type of water. Comment on the differences between the two water types.
(b) Suppose the $\mathrm{pH}$ of 7.10 in tap water was incorrectly recorded as $1.70 .$ How does this affect the mean? the median? What property of the median does this illustrate?

Trinity Steen

Numerade Educator

05:51

Problem 20

Reaction Time In an experiment conducted online at the University of Mississippi, study participants are asked to react to a stimulus. In one experiment, the participant must press a key upon seeing a blue screen. The time (in seconds) to press the key is measured. The same person is then asked to press a key upon seeing a red screen, again with the time to react measured. The table shows the results for six study participants. (FIGURE CAN'T COPY)
(a) Determine the mean, median, and mode reaction time for both blue and red.
(b) Does there appear to be a difference in the reaction time? What might account for any difference? How might this information be used?
(c) Suppose the reaction time of 0.841 for blue was incorrectly recorded as $8.41 .$ How does this affect the mean? the median? What property of the median does this illustrate?

Trinity Steen

Numerade Educator

04:40

Problem 21

Pulse Rates The following data represent the pulse rates (beats per minute) of nine students enrolled in a section of Sullivan's Introductory Statistics course. Treat the nine students as a population. (FIGURE CAN'T COPY)
(a) Determine the population mean pulse.
(b) Find three simple random samples of size 3 and determine the sample mean pulse of each sample.
(c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Julie Miller

Numerade Educator

03:53

Problem 22

Travel Time The following data represent the travel time (in minutes) to school for nine students enrolled in Sullivan's College Algebra course. Treat the nine students as a population. (FIGURE CAN'T COPY)
(a) Determine the population mean for travel time.
(b) Find three simple random samples of size 4 and determine the sample mean for travel time of each sample.
(c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to
a sample mean that equals the population mean?

Julie Miller

Numerade Educator

04:27

Problem 23

Carbon Dioxide Emissions The given data represent the fossil-fuel carbon dioxide $\left(\mathrm{CO}_{2}\right)$ emissions (in thousands of metric tons) of the top 10 emitters in 2007 . (FIGURE CAN'T COPY)
(a) Determine the mean $\mathrm{CO}_{2}$ emissions of the top 10 countries.
(b) Explain why the total emissions of a country is likely not the best gauge of $\mathrm{CO}_{2}$ emissions; instead, the emissions per capita (total emissions divided by population size) is the better gauge.
(c) Determine the mean and median per capita $\mathrm{CO}_{2}$ emissions of the top 10 countries. Which measure would an environmentalist likely use to support the position that per capita $\mathrm{CO}_{2}$ emissions are too high? Why?

Trinity Steen

Numerade Educator

06:43

Problem 24

Tour de Lance Lance Armstrong won the Tour de France seven consecutive years ( $1999-2005$ ). The following table gives the winning times, distances, speeds, and margin of victory. (FIGURE CAN'T COPY)
(a) Determine the mean and median of his winning times for the seven races.
(b) Determine the mean and median of the distances for the seven races.
(c) Determine the mean and median of his winning margin.
(d) Determine the mean winning speed by finding the mean of the data values in the table. Next, compute the mean winning speed by finding the total of the seven distances and dividing by the total of the seven winning times. Finally, compute the mean winning speed by dividing the mean distance by the mean winning time. Do the three values agree or are there differences?

Trinity Steen

Numerade Educator

00:50

Problem 25

Connection Time A histogram of the connection time, in seconds, to an Internet service provider for 30 randomly selected connections is shown. The mean connection time is 39.007 seconds and the median connection time is 39.065 seconds. Identify the shape of the distribution. Which measure of central tendency better describes the "center" of the distribution? (FIGURE CAN'T COPY)

Trinity Steen

Numerade Educator

00:42

Problem 26

Journal costs A histogram of the annual subscription cost (in dollars) for 26 biology journals is shown. The mean subscription cost is 1846 dollar and the median subscription cost is 1142 dollar. Identify the shape of the distribution. Which measure of central tendency better describes the "center" of the distribution? (FIGURE CAN'T COPY)

Trinity Steen

Numerade Educator

05:53

Problem 27

M\&Ms The following data represent the weights (in grams) of a simple random sample of $50 \mathrm{M} \& \mathrm{M}$ plain candies. (FIGURE CAN'T COPY) Determine the shape of the distribution of weights of M\&Ms by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the weight of a plain M\&M?

Julie Miller

Numerade Educator

05:07

Problem 28

Old Faithful We have all heard of the Old Faithful geyser in Yellowstone National Park. However, there is another, less famous, Old Faithful geyser in Calistoga, California. The following data represent the length of eruption (in seconds) for a random sample of eruptions of the California Old Faithful. (FIGURE CAN'T COPY)
Determine the shape of the distribution of length of eruption by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the length of eruption?

Julie Miller

Numerade Educator

05:23

Problem 29

Hours Working A random sample of 25 college students was asked, "How many hours per week typically do you work outside the home?" Their responses were as follows: (FIGURE CAN'T COPY)
Determine the shape of the distribution of hours worked by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes hours worked?

Julie Miller

Numerade Educator

05:21

Problem 30

A Dealer's Profit The following data represent the profit (in dollars) of a new-car dealer for a random sample of 40 sales. (FIGURE CAN'T COPY)
Determine the shape of the distribution of new-car profit by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the profit?

Julie Miller

Numerade Educator

01:08

Problem 31

Political Views A sample of 30 registered voters was surveyed in which the respondents were asked, "Do you consider your political views to be conservative, moderate, or liberal?" The results of the survey are shown in the table. (FIGURE CAN'T COPY)
(a) Determine the mode political view.
(b) Do you think it would be a good idea to rotate the choices conservative, moderate, or liberal in the question? Why?

Trinity Steen

Numerade Educator

01:06

Problem 32

Hospital Admissions The following data represent the diagnosis of a random sample of 20 patients admitted to a hospital. Determine the mode diagnosis. (FIGURE CAN'T COPY)

Trinity Steen

Numerade Educator

05:56

Problem 33

Resistance and Sample Size Each of the following three data sets represents the IQ scores of a random sample of adults. IQ scores are known to have a mean and median of 100. For each data set, determine the mean and median. For each data set recalculate the mean and median, assuming that the individual whose IQ is 106 is accidentally recorded as $160 .$ For each sample size, state what happens to the mean and the median. Comment on the role the number of observations plays in resistance. (FIGURE CAN'T COPY)

Trinity Steen

Numerade Educator

01:07

Problem 34

Mr. Zuro finds the mean height of all 14 students in his statistics class to be 68.0 inches. Just as Mr. Zuro finishes explaining how to get the mean, Danielle walks in late. Danielle is 65 inches tall. What is the mean height of the 15 students in the class?

James Kiss

Numerade Educator

02:16

Problem 35

Missing Exam Grade A professor has recorded exam grades for 20 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 19 readable scores is $84,$ what is the value of the unreadable score?

Christopher Stanley

Numerade Educator

02:04

Problem 36

Survival Rates Unfortunately, a friend of yours has been diagnosed with cancer. You obtain a histogram of the survival time (in months) of patients diagnosed with this form of cancer as shown in the figure. The median survival time for individuals with this form of cancer is 11 months, while the mean survival time is 69 months. What words of encouragement should you share with your friend from a statistical point of view? (FIGURE CAN'T COPY)

Julie Miller

Numerade Educator

00:37

Problem 37

Sullivan Survey Choose any quantitative variable from the Sullivan Survey. Draw a histogram of the data. Determine the mean and the median. Based on the mean, median, and histogram, describe the shape of the distribution. Which measure of central tendency better describes the "center" of the distribution?

Amrita Bhasin

Numerade Educator

02:21

Problem 38

Sullivan Survey Choose any quantitative variable from the Sullivan Survey. Now choose a qualitative variable, such as gender or political philosophy. Determine the mean and the median by the qualitative variable chosen. For example, if you chose gender as the qualitative variable, determine the mean and median by gender. Does there appear to be any difference in the measure of central tendency for each level of the qualitative variable?

Banhishikha Sinha

Numerade Educator

08:10

Problem 39

Linear Transformations Benjamin owns a small Internet business. Besides himself, he employs nine other people. The salaries earned by the employees are given next in thousands of dollars (Benjamin's salary is the largest, of course): $$30,30,45,50,50,50,55,55,60,75$$
(a) Determine the mean, median, and mode for salary.
(b) Business has been good! As a result, Benjamin has a total of 25,000 dollar in bonus pay to distribute to his employees. One option for distributing bonuses is to give each employee (including himself) 2500 dollar. Add the bonuses under this plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?
(c) As a second option, Benjamin can give each employee a bonus of $5 \%$ of his or her original salary. Add the bonuses under this second plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?
(d) As a third option, Benjamin decides not to give his employees a bonus at all. Instead, he keeps the 25,000 dollar for himself. Use this plan to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?

Trinity Steen

Numerade Educator

02:26

Problem 40

Linear Transformations Use the five test scores of 65,70,71 $75,$ and 95 to answer the following questions:
(a) Find the sample mean.
(b) Find the median.
(c) Which measure of central tendency best describes the typical test score?
(d) Suppose the professor decides to curve the exam by adding 4 points to each test score. Compute the sample mean based on the adjusted scores.
(e) Compare the unadjusted test score mean with the curved test score mean. What effect did adding 4 to each score have on the mean?

Jerelyn Nevil

Numerade Educator

02:18

Problem 41

Trimmed Mean Another measure of central tendency is the trimmed mean. It is computed by determining the mean of a data set after deleting the smallest and largest observed values. Compute the trimmed mean for the data in Problem $27 .$ Is the trimmed mean resistant? Explain.

Trinity Steen

Numerade Educator

01:06

Problem 42

Midrange The midrange is also a measure of central tendency. It is computed by adding the smallest and largest observed values of a data set and dividing the result by $2 ;$ that is, Midrange $=\frac{\text { largest data value }+\text { smallest data value }}{2}$ Compute the midrange for the data in Problem $27 .$ Is the midrange resistant? Explain.

Trinity Steen

Numerade Educator

04:34

Problem 43

Putting It Together: Shape, Mean and Median As part of a semester project in a statistics course, Carlos surveyed a sample of 50 high school students and asked, "How many days in the past week have you consumed an alcoholic beverage?" The results of the survey are shown next. $$\begin{array}{cccccccccc}
0 & 0 & 1 & 4 & 1 & 1 & 1 & 5 & 1 & 3 \\ \hline 0 & 1 & 0 & 1 & 0 & 4 & 0 & 1 & 0 & 1 \\ \hline 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ \hline 1 & 0 & 2 & 0 & 0 & 0 & 1 & 2 & 1 & 1 \\ \hline 2 & 0 & 1 & 0 & 1 & 3 & 1 & 1 & 0 & 3 \end{array}$$
(a) Is this data discrete or continuous?
(b) Draw a histogram of the data and describe its shape.
(c) Based on the shape of the histogram, do you expect the mean to be more than, equal to, or less than the median?
(d) Determine the mean and the median. What does this tell you?
(e) Determine the mode.
(f) Do you believe that Carlos's survey suffers from sampling bias? Why?

James Kiss

Numerade Educator

03:02

Problem 44

44. FICO Scores The Fair Isaacs Corporation has devised a model that is used to determine creditworthiness of individuals, called a FICO score. FICO scores range in value from 300 to 850 with a higher score indicating a more credit-worthy individual. The distribution of FICO scores is skewed left with a median score of 723.
(a) Do you think the mean FICO score is greater than, less than, or equal to $723 ?$ Justify your response.
(b) What proportion of individuals have a FICO score above $723 ?$

Julie Miller

Numerade Educator

01:46

Problem 45

Why is the median resistant, but the mean is not?

Christopher Stanley

Numerade Educator

01:11

Problem 46

A researcher with the Department of Energy wants to determine the mean natural gas bill of households throughout the United States. He knows the mean natural gas bill of households for each state, so he adds together these 50 values and divides by 50 to arrive at his estimate. Is this a valid approach? Why or why not?

Christopher Stanley

Numerade Educator

02:24

Problem 47

47. Net Worth According to the Statistical Abstract of the United States, the mean net worth of all households in the United States in 2007 was 556,300 dollar, while the median net worth was 120,300 dollar.
(a) Which measure do you believe better describes the typical U.S. household's net worth? Support your opinion.
(b) What shape would you expect the distribution of net worth to have? Why?
(c) What do you think causes the disparity in the two measures of central tendency?

Julie Miller

Numerade Educator

01:31

Problem 48

You are negotiating a contract for the Players Association of the NBA. Which measure of central tendency will you use to support your claim that the average player's salary needs to be increased? Why? As the chief negotiator for the owners, which measure would you use to refute the claim made by the Players Association?

Christopher Stanley

Numerade Educator

01:29

Problem 49

In January 2011 , the mean amount of money lost per visitor to a local riverboat casino was $\$ 135 .$ Do you think the median was more than, less than, or equal to this amount? Why?

Christopher Stanley

Numerade Educator

02:01

Problem 50

For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning.
(a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2011
(b) Most popular major for students enrolled in a statistics course
(c) Average test score when the scores are distributed symmetrically
(d) Average test score when the scores are skewed right
(e) Average income of a player in the National Football League
(f) Most requested song at a radio station

Christopher Stanley

Numerade Educator