(2.1) Relative Frequency pg. 56
Relative frequency5 f n
i
(2.2) Class Width pg. 60
W Largest value Smallest value Number of classses
Key Terms
All-inclusive classes pg. 59
Bar chart pg. 74
Class boundaries pg. 60
Class width pg. 60
Continuous data pg. 57
Cumulative frequency distribution pg. 61
Cumulative relative frequency distribution pg. 61
Dependent variable pg. 86
Discrete data pg. 53
Equal-width classes pg. 59
Frequency distribution
pg. 53
Frequency histogram pg. 62
Independent variable pg. 86
Line chart pg. 83
Mutually exclusive classes pg. 59
Ogive pg. 65
Pie chart pg. 77
Relative frequency pg. 55
Scatter diagram or scatter plot pg. 86
Chapter Exercises
Conceptual Questions2-62. Discuss the advantages of constructing a relative frequency distribution as opposed to a frequency distribution.
2-63. What are the characteristics of a data set that would lead you to construct a bar chart?
2-64. What are the characteristics of a data set that would lead you to construct a pie chart?
2-65. State the differences between a line chart and a scatter plot.
Business Applications
2-66. As it has become more popular to stream movies over the Internet at home from such sources as Hulu and Netflix, ticket sales at movie theaters have been affected.
The following data reflect annual ticket sales in billions
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Graphs, Charts, and Tables—Describing Your Data$10: youths 6 to 12 years old
$15: students 13 to 17 years old
$20: adult 18+ years old
Group the age into each category and determine a pie chart for the management.
d. Referring to part c, construct a bar chart to depict the proportions for the management.
2-69. Dave is doing a study on recycling. He visited the civic amenity site (CA site) and household waste recycling center (HWRC) where the public can dispose of household waste to collect his data.
Following are the amount of waste (in tons) on three different items most collected by the two facilities in a month:
Compost Paper and Card Glass
CA site 497 427 330
HWRC 275 386 154
a. Help Dave to construct an appropriate graph for the data he collected.
b. In a brief report for Dave, describe the observations on the graph drawn in part a.
2-70. Catherine just graduated and received a Bachelor’s Degree. She wants to determine whether she needs to continue pursuing a Master’s Degree or start working.
She did a study on the monthly income earned by a person based on the number of years they spent studying after high school. The data collected from a sample of 18 are as follows:
Years Income ($) Years Income ($)
3 3,450 7 6,560
3 4,560 10 9,890
5 5,650 9 10,640
8 7,500 4 5,670
6 8,100 3 4,120
4 4,650 5 5,480
5 6,750 3 3,850
6 7,340 4 4,980
8 8,540 7 6,880
a. Construct an appropriate graph for the data collected by Catherine.
b. Determine and describe the relationship between the number of years spent in study after high school and the income earned.
2-71. A computer software company has been looking at the amount of time customers spend on hold after their call is answered by the central switchboard. The company would like to have at most 2% of the callers wait two minutes or more. The company’s calling service has provided the following data showing how long each of last month’s callers spent on hold:
at U.S. movie theaters between 1995 and 2015. Develop an appropriate chart to describe the trend in ticket sales over this time period.
Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Tickets
(billions)
1.22 1.27 1.42 1.45 1.44 1.39 1.44 1.58 1.55 1.47
Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Tickets
(billions)
1.39 1.41 1.4 1.39 1.42 1.33 1.28 1.39 1.34 1.27 1.28
Source: www.the-numbers.com/market/
2-67. Barbara is going to Tokyo for a business trip that will last for a period of 10 days. To make sure she brings enough clothes for the trip, she checks the weather report to estimate the general weather pattern during her stay in Tokyo. The following are the estimated daily average temperatures she found from a website:
Day Temperature (°C)
1 12
2 13
3 14
4 17
5 16
6 10
7 13
8 11
9 10
10 7
a. Develop an appropriate graph that will determine the movement of the temperatures.
b. Describe your observations on the movement of the temperature. What would your recommendation be for Barbara?
2-68. A museum is planning to adjust the price of its entry tickets. The management recorded a group of visitors’
ages for a specific day. The sample collected by the museum is as follows:
34 24 14 23 56 32 54 65 47 23
54 12 35 46 7 35 24 34 34 63
25 35 14 13 35 46 64 24 35 42
24 35 63 21 33 19 53 10 9 59
14 11 53 63 25 62 23 32 13 35
a. Develop a frequency distribution and histogram for the management.
b. Construct a stem and leaf diagram for the management.
c. If the museum would like to categorize its tickets into the following categories:
Free: children under 5 years old and seniors 60 years old and above
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Case 2.1 Server Downtime
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Chapter 2 95Class Number
Less than 15 seconds 456
15 to less than 30 seconds 718 30 to less than 45 seconds 891 45 to less than 60 seconds 823 60 to less than 75 seconds 610 75 to less than 90 seconds 449 90 to less than 105 seconds 385 105 to less than 120 seconds 221 120 to less than 150 seconds 158 150 to less than 180 seconds 124 180 to less than 240 seconds 87
More than 240 seconds 153
a. Develop a relative frequency distribution and ogive for these data.
b. The company estimates it loses an average of $30 in business from callers who must wait two minutes or more before receiving assistance. The company thinks that last month’s distribution of waiting times is typical. Estimate how much money the company is losing in business per month because people have to wait too long before receiving assistance.
2-72. DLT is an authorized dealer for selling desktops, laptops, tablets, and other computer accessories for certain computer technology companies. The following table shows the sales of DLT’s three best-selling products in the past six months:
January February March April May June
Desktops 77 45 23 19 12 8
Laptops 31 25 34 51 44 53
Tablets 10 17 19 32 59 67
a. Develop an appropriate chart for the above data.
b. Based on the above chart, write a brief report for DLT.
Computer Software Exercises
2-73. The file titled Diesel$ contains the average on-highway diesel prices for the last Friday of the month in the years 2013–2015. Develop a line chart that shows the trend in diesel prices, and write a short statement that describes the price trend.
2-74. The data in the file titled Digital provide the brand of digital devices owned by a sample of consumers.
Produce a pie chart that represents the market shares obtained from the referenced sample. Indicate the market shares and the identity of those
manufacturers in the pie chart.
2-75. The file Home-Prices contains information about single-family housing prices in 100 metropolitan areas in the United States.
a. Construct a frequency distribution of median single-family home prices. Use the 2k Ú n guideline to determine the appropriate number of classes.
b. Construct a cumulative relative frequency distribution for median single-family home prices.
2-76. Elliel’s Department Store tracks its inventory on a monthly basis. Monthly data for 2008 through 2012 are in the file called Elliels.
a. Construct a line chart showing the monthly inventory over the five years. Discuss what this graph implies about inventory.
b. Sum the monthly inventory figures for each year.
Then present the sums in bar chart form. Discuss whether you think this is an appropriate graph to describe the inventory situation at Elliels.
2-77. The file titled Diesel$ contains the average nationwide diesel price on the last Friday of the month for the years 2013–2015.
a. Construct a histogram with 11 classes.
b. What are the possible reasons why one of the classes has a frequency count of zero?
Case 2.1 Server Downtime
After getting outstanding grades in high school and scoring very high on his ACT and SAT tests, Clayton Haney had his choice of colleges but wanted to follow his parents’ legacy and enrolled at Northwestern University. Clayton soon learned that there is a big difference between getting high grades in high school and being a good student. Although he was recognized as being quite bright and very quick to pick up on things, he had never learned how to study. As a result, after slightly more than two years at Northwestern, Clayton was asked to try his luck at another uni- versity. To the chagrin of his parents, Clayton decided that college was not for him.
After short stints working for a computer manufacturer and as a manager at a convenience store, Clayton landed an entry-level job working for EDS. EDS contracts to support information tech- nology implementation and application for companies in the
United States and throughout the world. Clayton received training in virtually all aspects of personal computers and local area net- works and was assigned to work for a client in the Chicago area.
Clayton’s first assignment was to research the downtime on one of the client’s primary network servers. He was asked to study the downtime data for the month of April and to make a short pres- entation to the company’s management. The downtime data are in a file called Server Downtime. Although Clayton is very good at solving computer problems, he has had no training or experience in analyzing data, so he is going to need some help.
Required Tasks:
1. Construct a frequency distribution showing the number of times during the month that the server was down for each downtime cause category.
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Graphs, Charts, and Tables—Describing Your DataCase 2.2 Hudson Valley Apples, Inc.
As a rule, Stacey Fredrick preferred to work in the field rather than do “office” work in her capacity as a midlevel manager with Hudson Valley Apples, Inc., a large grower and pro- cessor of apples in the state of New York. However, after just leav- ing a staff meeting at which she was asked to prepare a report of apple consumption in the United States, Stacey was actually look- ing forward to spending some time at her computer “crunching some numbers.” Arden Golchein, senior marketing manager, indi- cated that he would e-mail her a data file that contained apple consumption data from 1977 through 2015 and told her that he wanted a report using graphs, charts, and tables to describe apple consumption.
When she got to her desk, the e-mail was waiting, and she saved the file under the name Hudson Valley Apples. Stacey had done quite a bit of descriptive analysis in her previous job with the New York State Department of Agriculture, so she had
several ideas for types of graphs and tables that she might con- struct. She began by creating a list of the tasks that she thought would be needed.
Required Tasks:
1. Construct a line chart showing the total annual availability of apples.
2. Construct one line chart that shows two things: the annual availability of fresh apples and the annual availability of processed apples.
3. Construct a line chart that shows the annual availability for each type of processed apples.
4. Construct a histogram for the total annual availability of apples.
5. Write a short report that discusses the historical pattern of apple availability. The report will include all pertinent charts and graphs.
Case 2.3 Pine River Lumber Company—Part 1
Martin Bledsoe wears several hats at the Pine River Lumber Company, including process improvement team leader, shipping manager, and assistant human resources manager. Pine River Lumber makes cedar fencing materials at its Naples, Idaho, facility, employing about 160 people.
More than 75% of the cost of the finished cedar fence boards is in the cedar logs that the company buys on the open market. There- fore, it is very important that the company get as much finished product as possible from each log. One of the most important steps in the manufacturing process is referred to as the head rig. The head rig is a large saw that breaks down the logs into slabs and cants. Figure C-2.3-A shows the concept. From small logs with diameters of 12 inches or less, one cant and four or fewer usable slabs are obtained. From larger logs, multiple cants and four slabs are obtained. Finished fence boards can be produced from both the slabs and the cants.
At some companies, the head rig cutting operation is auto- mated and the cuts are made based on a scanner system and com- puter algorithms. However, at Pine River Lumber, the head rig is operated manually by operators who must look at a log as it arrives and determine how best to break the log down to get the most fin- ished product. In addition, the operators are responsible for mak- ing sure that the cants are “centered” so that maximum product can be gained from them.
Recently, Martin Bledsoe headed up a study in which he vid- eotaped 365 logs being broken down by the head rig. All three
operators, April, Sid, and Jim, were involved. Each log was marked as to its true diameter. Then Martin observed the way the log was broken down and the degree to which the cants were properly centered. He then determined the projected value of the finished product from each log given the way it was actually cut.
In addition, he determined what the value would have been had the log been cut in the optimal way. Data for this study are in a file called Pine River.
You have been asked to assist Martin by analyzing these data using graphs, charts, and tables as appropriate. He wishes to focus on the lost profit to the company and whether there are differences among the operators. Also, do the operators tend to do a better job on small logs than on large logs? In general, he is hoping to learn as much as possible from this study and needs your help with the analysis.
FIGURE C-2.3-A Log Breakdown at the Head Rig
Slabs
Slabs
Cant
2. Develop a bar chart that displays the data from the frequency distribution in part a.
3. Develop a histogram that displays the downtime data.
4. Develop a pie chart that breaks down the percentage of total downtime that is attributed to each downtime cause during the month.
5. Prepare a short written report that discusses the downtime data. Make sure you include the graphs and charts in the report.
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WHY YOU NEED TO KNOW
Suppose you are the advertising manager for a major airline and you want to develop an ad campaign touting how much cheaper your fares are than the competition’s. You must be careful that your claims are valid. First, the Federal Trade Commission (FTC) is charged with regulating advertising and requires that advertising be truthful. Second, customers who can show that they were misled by an incorrect claim about prices could sue your company. You need to use statistical procedures to determine the validity of any claim you might want to make about your prices. Graphs and charts provide effective tools for transforming data into information; however, they are only a starting point. Graphs and charts do not reveal all the
Measures of Center and Location (pg. 98–118)
o u t c o m e 1 Compute the mean, median, mode, and weighted mean for a set of data and use these measures to describe data.
o u t c o m e 2 Construct a box and whisker graph and interpret it.
Measures of Variation
(pg. 119–129)
o u t c o m e 3 Compute the range, interquartile range, variance, and standard deviation and use these measures to describe data.
Using the Mean and Standard Deviation Together (pg. 130–137)
o u t c o m e 4 Compute a z-score and the coefficient of variation and apply them in decision-making situations.
o u t c o m e 5 Use the Empirical Rule and Tchebysheff’s theorem.
3.1
3.2
3.3
97
Describing Data Using Numerical Measures
3
Review the definitions for nominal, ordinal, interval, and ratio data in Section 1.4.
Examine the statistical software, such as Excel, used during this course to identify the tools for computing descriptive measures.
For instance, in Excel, look at the function wizard and the descriptive statistics tools on the Data tab under Data Analysis.
Review the material on frequency histograms in Section 2.1, paying special attention to how histograms help determine where the data are centered and how the data are spread around the center.
Quick Prep
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Describing Data Using Numerical Measuresinformation contained in a set of data. To make your descriptive toolkit complete, you need to become familiar with key descriptive measures that are widely used to fully describe data.
You will need to combine the graphical tools discussed in Chapter 2 with the numerical measures introduced in this chapter.