# 5.7: Chapter 5 Review

**At Grade**Created by: Bruce DeWItt

### Chapter 5 Summary

In this chapter, we have learned that when working with a set of data it is important to choose an appropriate type of graphical display so that we can see what the data looks like. Bar graphs and pie charts are useful ways to display categorical data. Time plots are line graphs that help us to see how a given variable has changed over a specified period of time. And, when working with numerical data, we have learned how to make dot plots, stem plots, histograms, and box plots. It is also possible to make graphs so that comparisons can be made between more than one data set. Back-to-back stem plots and parallel box plots are two such types of graphs.

The next step is to analyze the data set(s) by calculating numerical statistics. The statistics that give us an idea of where the center of the data is are the mean, median, and mode. These statistics are measures of central tendency, and give us an idea of where the 'average' of the data lies. The range, inner quartile range (IQR), and the standard deviation are all measures of the spread of a set of data. We have also learned how to calculate the five number summary, which divides a set of data into quarters and allows us to construct a box plot.

Once the graphs are constructed and the statistics are calculated, we have learned to describe what these show. When describing a numerical set of data, in addition to explaining where the center and spread are, we also describe the shape of the distribution and whether or not any outliers are present in the data. The shapes that we focused on are symmetrical distributions and skewed distributions, remembering that the direction of the skew is toward the tail or outliers. We learned to make appropriate conclusions and comparisons that are based on the data, graphs, and statistics. Statisticians should avoid opinions and judgment statements as much as possible.

We learned that the 1.5*(IQR) Criterion can be used to determine whether or not any data values are outliers. And, that the mean and standard deviation are easily changed, so these statistics are not the appropriate measures of center and spread when working with data that contains outliers or is skewed.

### Chapter 5 Review Exercises

1) Multiple-Choice: Which of the following can be inferred from this histogram?

a) The mode is 1.

b) mean < median

c) median < mean

d) The distribution is skewed left.

e) None of the above can be inferred from this histogram.

2) The owner of a small company is trying to determine whether he should go with a different company for his shipping needs. He needs to analyze the weights of the packages that his company ships out. This graph shows the distribution of the weight of packages that were shipped during the last month.

a) Calculate the mean, standard deviation, mode, and range for this data. Use a calculator for mean and standard deviation.

b) Determine the five number summary for this data. Construct a box plot for this data.

c) Which of these two graphs is more informative? Explain.

d) He figures that he will save money with the new company on any packages that weigh less than 16.75 ounces. What percent of packages weighed more than 16.75 ounces?

3) After some bullying issues were brought to light in a big high school, a committee was formed to study the issue. A questionnaire was designed that contained several questions related to bullying and safety. A stratified random sample was selected that included students from all four grade levels. The table that follows shows the responses to one of the questions on the questionnaire.

a) Use Excel or Google Docs to create a pie chart that shows the results of this survey question. Be sure to include labels, percents, a title, and a key if needed.

b) Describe what the graph shows in context. Be sure to include percents to support your observations.

c) Comment on whether or not the committee should be concerned. Explain.

In questions 4-7, match the distribution with the choice of the correct real-world situation that best fits the graph.

4)

5)

6)

7)

a) Andy collected and graphed the heights of all the \begin{align*}12^{th}\end{align*} grade students in his high school.

b) Brittany asked each of the students in her statistics class to bring in 20 pennies selected at random from their pocket or piggy bank. She created a plot of the dates of the pennies.

c) Maya asked her friends what their favorite movie was this year and graphed the results.

d) Jeno bought a large box of doughnut holes at the local pastry shop, weighed each of them, and then plotted their weights to the nearest tenth of a gram.

Questions 8 - 17 are multiple-choice questions. Select the best answer from the choices given.

8) Which of the following box plots matches the histogram?

9) Identify the 5 number summary for this set of numbers:

12,356; 16,564; 15,684; 12,358; 15,987; 13,556; 18,564; 18,965; 19,683; 18,432; 18,563; 19,352

a) {12,356; 14,600; 17,498; 18,000; 19,683}

b) {12,356; 14,620; 17,498; 18,764.5; 19683}

c) {12,356; 14,650.5; 17,498; 18,700.5; 19683}

d) {12,356; 14,683; 17,500; 18,800; 19683}

e) {12,356; 14,695.5; 17,900; 18,888; 19683}

10) Thirty students took a statistics examination having a maximum of 50 points. The grade distribution is given in the following stem-and-leaf plot:

The median grade is equal to:

a) 30.5

b) 30.0

c) 25.0

d) 28.5

e) 44.0

11) Ms. Davis conducted a survey of the 44 students in her stats classes and asked how tall each student is in inches. Here is the five-number summary of the students’ data:

\begin{align*}\left \{57, 64, 67, 69, 79 \right \}\end{align*}

Approximately how many people are shorter than 64 inches tall?

a) 8

b) 21

c) 22

d) 11

e) 18

12) In which scenario(s) would it be better to use the 5-number summary versus the mean and standard deviation?

a) a graph that is skewed

b) a graph that is fairly symmetric

c) a graph that is symmetric but has several high outliers

d) Both choice (b) and (c)

e) Both choice (a) and (c)

f) All of (a), (b) and (c)

13) Suppose the lowest score on an English exam was 35% and the highest score was 90%. If the teacher of the class was to examine her students' test scores, which type of distribution would she prefer to see? One that is…

a) skewed to the right

b) skewed to the left

c) fairly symmetric

d) none of the above

14) Several people were surveyed as they were leaving a movie theatre. Among other things, they were asked how much money they had spent. They answers were: $14, $17.50, $16, $16, $19.25, $12.75, $16, $37.75, $13.50 and $17. It was later discovered that the person who answered “$37.75” actually spent $17.75. Which of the following would **not** change as a result?

a) the box plot

b) the mean & the mode

c) the median & the mode

d) the standard deviation

e) they all change

15) What does the following five-number summary tell you about the shape of the distribution? \begin{align*}\left \{ 5, 7.7, 9, 10.9, 24\right \}\end{align*}

a) skewed to the right

b) skewed to the left

c) symmetric

d) uniform

e) cannot determine

16) According to the 1.5*(IQR) Criterion, what are the two cut-off values for determining whether the data set in question #15 contains any outliers?

a) 5 & 24

b) 7.7 & 10.9

c) 11.3 & 29.9

d) 4.5 & 14.1

e) 2.9 & 15.7

17) A class survey was conducted to determine students’ preferences. One question regarded favorite sport to watch on TV. The results are as follows: 9 said “football”; 12 said “hockey”; 5 said “basketball”; 6 said “baseball” and 3 said “other”. What would the central angle be for “hockey” in a pie chart of this data?

a) 65°

b) 123°

c) 90°

d) 34°

e) 111°

18) The following two graphs are based from the US Census Bureau, 2008 ('per capita' means per person). The dots represent actual data values, and the red curves represent models that can be used to predict future trends. Study the two graphs and answer the questions that follow.

Source:http://www.rationalfuturist.com August 2, 2011.

a) What type of graphs are these?

b) Describe the trend that each graph shows separately. This should be in context.

c) Notice that the horizontal scales are the same. Compare and contrast the trends that are shown in the two different graphs in context.

d) Approximately how many cell phones were there per person in 1997? In 2005? How many will there be, if the trend continues as the model indicates, in 2018?

e) Approximately how many landlines were there per person at the peak? What year did this occur? If the trend continues as the model indicates, how many landlines will there be per person in 2015?

19) The AHS Tornadoes and the BHS Bengals are big rivals! Every year students try to prove that their school is better at sports than the other school. The table below shows the number of points scored by each school's basketball team during the last 15 games played.

Tornadoes |
Bengals |

58 |
74 |

90 |
81 |

71 |
73 |

64 |
63 |

58 |
58 |

63 |
84 |

60 |
92 |

72 |
38 |

48 |
77 |

59 |
84 |

72 |
95 |

62 |
66 |

57 |
70 |

64 |
68 |

49 |
72 |

a) Construct a back-to-back stem plot for the data.

b) Calculate the five number summary, mean and standard deviation for both teams.

c) Construct parallel box plots for the data.

d) Compare the two distributions. This should be done in context and include at least three distinct comparisons.

e) What other information would you like to know when comparing these two basketball teams? Explain.

20) The table that follows shows the percent of people, 25 years and older, who are high school graduates for several states in the central United States. According to the 2010 U. S. Census website.

Source:http://quickfacts.census.gov

a) Construct a histogram (use Xmin = 79.5%, and bin width = 1.5%).

b) Calculate the five number summary.

c) Identify any outliers. Use the outlier test.

d) Accurately sketch a box plot.

e) What is the range? The IQR? The mode?

f) Calculate the mean and standard deviation.

g) Compare the mean and the median.

(i.e. which is larger? How different are they?)

h) In this case would the 5#-summary or the mean & standard deviation be more appropriate? Why?

i) Describe the distribution. Be thorough! Don't forget your S.O.C.C.S! (shape, outliers, center, context, & spread)

j) According to the Census data, where does Minnesota fall?

21) An employer in Minneapolis was interested in determining how much money his employees were spending on parking each week. An SRS of 50 employees was selected to complete a questionnaire about parking. Several questions were asked including where they park, how much they spend per week, how often they have difficulties finding spots, if they pay daily, weekly, or monthly, etc. The following table is the average weekly expenditure for parking for this sample of 50 employees.

a) Construct a split-stem plot

b) Calculate the five number summary.

c) Identify any outliers. Use the outlier test

d) Accurately sketch a box plot. (to scale with labels)

e) What is the range? The IQR? The mode?

f) Calculate the mean and standard deviation.

g) Compare the mean and the median.

h) In this case would the 5#-summary or the mean & standard deviation be more appropriate? Why?

i) Describe the distribution. Be thorough! Remember your S.O.C.C.S!

#### Image References:

Gasoline.http://www.education.vic.gov.au

Cars.http://www.icoachmath.com

School Lunch pictogram. http://alex.state.al.us

Dot plot.http://cwx.prenhall.com

Stem plot example.http://www.basic-mathematics.com

Shapes of distributions.http://thesocietypages.org

Weight of Jessica graph. http://www.stat.psu.edu

Crowded stem plot. http://illuminations.nctm.org/

Three histogram example. http://classes.cec.wustl.edu

Package weight graph. http://flylib.com