CH. 2 Notes Review Complete Solution
Section 2.1
Organize Qualitative Data in Tables
When qualitative data are collected, we often first determine the
... [Show More] number of individuals within each category.
A frequency distribution lists each category of data and the number of occurrences for each category of data.
Ex1) Organizing Qualitative Data into a Frequency Distribution
Problem: A physical therapist wants to determine types of rehabilitation required by her patients. To do so, she obtains a simple random sample of 30 of her patients and records the body part requiring rehabilitation. See Table 1. Construct a frequency distribution of location of injury.
TABLE 1
Back Back Hand
Wrist Back Groin
Elbow Back Back
Back Shoulder Shoulder
Hip Knee Hip
Neck Knee Knee
Shoulder Shoulder Back
Back Back Back
Knee Knee Back
Hand Back Wrist
Data from Krystal Catton, student at Joliet Junior College
Solution: The first thing we need to do is come up with a list of the different body parts that were treated.
We have back, hand, wrist, groin, elbow, shoulder, hip, knee, and neck.
So we list those nine body parts in a table and tally the number of occurrences of each body part; like so,
Body Part Tally
Back ||||| ||||| || 12
Hand || 2
Wrist || 2
Groin | 1
Elbow | 1
Shoulder |||| 4
Hip || 2
Knee ||||| 5
Neck | 1
Body Part Frequency
Back 12
Hand 2
Wrist 2
Groin 1
Elbow 1
Shoulder 4
Hip 2
Knee 5
Neck 1
In any frequency distribution, it is a good idea to add up the frequency column to make sure that it equals the number of observations.
In Example 1, the frequency column totals to 30 as it should because there are 30 body parts (observations).
Often, we want to know the relative frequency of the categories rather than the frequency.
The relative frequency is the proportion (or percent) of observations within a category and is found using the formula
Relative frequency=Frequency/(Sum of all frequencies)
A relative frequency distribution lists each category of data together with the relative frequency.
Ex2) Constructing a Relative Frequency Distribution of Qualitative Data
Problem: Using the summarized data in Table 2, construct a relative frequency distribution.
TABLE 2
Body Part Tally Frequency
Back |||||\|||||\|| 12
Wrist || 2
Elbow | 1
Hip || 2
Shoulder |||| 4
Knee ||||\ 5
Hand || 2
Groin | 1
Neck | 1
Approach: Add all the frequencies and then use
Relative frequency=Frequency/(Sum of all frequencies)
to compute the relative frequency of each category of data.
Solution: The sum of all the values in the frequency column in Table 2 is 30. We now compute the relative frequency of each category. For example, the relative frequency of the category Back is 12/30 = 0.4. The relative frequencies are shown in column 3 of Table 3. From the distribution, the most common body part for rehabilitation is the back.
TABLE 3:Relative Frequency Distribution
Body Part Frequency Relative Frequency
Back 12 12/30 = 0.4
Wrist 2 2/30 ≈0.0667
Elbow 1 1/30 ≈0.0333
Hip 2 2/30 ≈0.0667
Shoulder 4 4/30 ≈0.1333
Knee 5 5/30 ≈0.1667
Hand 2 2/30 ≈0.0667
Groin 1 1/30 ≈0.0333
Neck 1 1/30 ≈0.0333
Total 30 1
Construct Bar Graphs
A common device for graphically representing qualitative data is a bar graph.
A bar graph is constructed by labeling each category of data on either the horizontal or vertical axis and the frequency or relative frequency of the category on the other axis. Rectangles of equal width are drawn for each category. The height of each rectangle represents the category's frequency or relative frequency.
Ex3) Constructing a Frequency and Relative Frequency Bar Graph
Problem: Use the data summarized in Table 3 to construct a frequency bar graph and relative frequency bar graph.
Approach: Use a horizontal axis to indicate the categories of the data (body parts) and a vertical axis to represent the frequency or relative frequency. Draw rectangles of equal width to the height that is the frequency or relative frequency for each category. The bars do not touch each other.
Solution:
In bar graphs, the order of the categories does not usually matter. However, bar graphs that have categories arranged in decreasing order of frequency help prioritize information for decision-making purposes.
A Pareto chart is a bar graph whose bars are drawn in decreasing order of frequency or relative frequency.
Side-by-Side Bar Graphs
Suppose we want to know whether more people finished college in 2012 than in 1990.
We could draw a side-by-side bar graph to compare the data for the two different years.
When comparing data sets, it is best to use relative frequencies because different sample or population sizes make comparisons using frequencies difficult or misleading.
Figure 3
Ex4) The frequency data in Table 4 represent the educational attainment (level of education) in 1990 and 2012 of adults 25 years and older who are U.S. residents. The data are in thousands. So 39,344 represents 39,344,000.
TABLE 4
Educational Attainment 1990 2012
Not a high school graduate 39,344 25,276
High school diploma 47,643 62,113
Some college, no degree 29,780 34,163
Associate's degree 9,792 19,737
Bachelor's degree 20,833 40,561
Graduate or professional degree 11,478 22,730
Totals 158,870 204,580
Data from U.S. Census Bureau
(a) Draw a side-by-side bar graph
Horizontal Bars
So far we have only looked at bar graphs with vertical bars. However, the bars may also be horizontal. Horizontal bars are preferable when category names are lengthy. For example, Figure 4 uses horizontal bars to display the same data as in Figure 3.
Figure 4
Construct Pie Charts
Pie charts are typically used to present the relative frequency of qualitative data. In most cases, the data are nominal, but ordinal data can also be displayed in a pie chart.
A pie chart is a circle divided into sectors. Each sector represents a category of data. The area of each sector is proportional to the frequency of the category.
Decide between drawing a bar graph or pie chart
Pie Charts are useful for showing the division of all possible values of a qualitative variable into its parts. However, because angles are often hard to judge in pie charts, they are not as useful in comparing two specific values of the qualitative variable. Instead, the emphasis is on comparing the part to the whole.
Bar Graphs are useful when we want to compare the different parts, not necessarily the parts to the whole.
Section 2.1 Interactive Assignment
2.1.24a A phlebotomist draws the blood of a random sample of 50 patients and determines their blood types as shown.
O A A O O B O B A O
AB AB A B AB A O A A O
AB O A B A A A A O A
O A O AB A O AB A A O
O O O O O O A O A O
Construct a frequency distribution.
Blood Type Frequency
A ||||\||||\||||\|||| 19
AB ||||\| 6
B |||| 4
O ||||\||||\||||\||||\| 21
2.1.24b A phlebotomist draws the blood of a random sample of 50 patients and determines their blood types as shown.
O O A O O B O B A O
AB AB A B B A O A A O
AB O A B A A A A O A
O A O B A O B A A O
O A O O O O A O A O
Blood Type Relative Frequency
A 19/50≈0.38
AB 3/50≈0.06
B 7/50≈0.14
O 21/50≈0.42
2.1.24c-h A phlebotomist draws the blood of a random sample of 50 patients and determines their blood types as shown. Complete parts (a) through (f).
O A A A O B O B A O
AB AB A B B A O O A O
AB O A B A A A A O A
O A O B A O B A A O
O A O A O O A O A O
Blood Type Frequency
A ||||\||||\||||\||||\| 21
AB ||| 3
B ||||\|| 7
O ||||\||||\||||\|||| 19
(a) According to the data, which blood type is most common?
A
(b) According to the data, which blood type is least common?
AB
(c) Use the results of the sample to conjecture the percentage of the population that has type O blood. Is this an example of descriptive or inferential statistics? Select the correct choice below and fill in the answer box to complete your choice.
38 %; inferential
(d) Contact a local hospital and ask them the percentage of the population that is blood type O. Why might the results differ?
The results might differ because there is always a chance that the sample surveyed is unlike the population.
(e) Draw a frequency bar graph.
(f) Draw a relative frequency bar graph.
2.1.17 Over the course of a decade, a certain police department issued 155.7 thousand speeding tickets. The ages of the males and females who received tickets are shown below. Use this information to answer parts a through c.
Age Groups Males
(in thousands) Females
(in thousands)
16-25 26.3 22.4
26-35 18.4 17.3
36-45 15.4 15.5
46-55 7.4 8.2
> 56 12.4 12.4
(a) Construct a relative frequency distribution of the males who received tickets. (Round answers to three decimal places.)
Add all of the frequencies for the male categories together to get the total number of males that received tickets. Divide the frequency of each male age group by the total number of tickets received by men to find the relative frequencies. 79.9
Age Group Relative Frequency
16-25 26.3 / 79.9 ≈ 0.329
26-35 18.4 / 79.9 ≈ 0.230
36-45 15.4 / 79.9 ≈ 0.193
46-55 7.4 / 79.9 ≈ 0.093
> 56 12.4 / 79.9 ≈ 0.155
(b) Construct a relative frequency distribution of the females who received tickets. (Round answers to three decimal places.)
Add all of the frequencies for the female categories together to get the total number of females that received tickets. Divide the frequency of each female age group by the total number of tickets received by women to find the relative frequencies. 75.8
Age Group Relative Frequency
16-25 22.4 / 75.8 ≈ 0.296
26-35 17.3 / 75.8 ≈ 0.228
36-45 15.5 / 75.8 ≈ 0.204
46-55 8.2 / 75.8 ≈ 0.108
> 56 12.4 / 75.8 ≈ 0.164
(c) Construct a side-by-side relative frequency bar graph. Choose the correct graph below where in each age grouping, the left bar represents males and the right bar represents females.
2.1.24i A phlebotomist draws the blood of a random sample of 50 patients and determines their blood types as shown.
O O A A O B O B A O
AB AB A B AB O O A A O
AB O A B A A A A O A
O A O AB A O AB A A O
O O O O O O A O A O
Draw a pie chart.
Section 2.1 Homework
2.1.2 A frequency distribution lists the number of occurrences of each category of data, while a relative frequency distribution lists the proportion of occurrences of each category of data.
2.1.3 In a relative frequency distribution, what should the relative frequencies add up to?
Select the correct choice and, if necessary, fill in the answer box to complete your choice.
The relative frequencies add up to 1.
2.1.4 What is a bar graph? What is a Pareto chart?
What is a bar graph?
A bar graph is a horizontal or vertical representation of the frequency or relative frequency of the categories. The height of each rectangle represents the category's frequency or relative frequency.
What is a Pareto chart?
A Pareto chart is a bar graph whose bars are drawn in decreasing order of frequency or relative frequency.
2.1.5 The pie chart below depicts the beverage size customers choose while at a fast food restaurant. Complete parts (a) through (c).
(a) What is the most popular size? What percentage of customers choose this size?
Small; 44%
(b) What is the least popular size? What percentage of customers choose this size?
Medium; 11%
(c) What percent of customers choose a large-sized beverage?
32%
2.1.7
2.1.9 The following data represent the percent of owners of an electronic device that plan to purchase a replacement device within the next 12 months based on a survey of 1100 adults in a certain country. Complete parts (a) through (c).
(a) What percent of game console owners plan to buy a replacement device within the next 12 months?
10.4%
(b) If there are 250 million individuals who own a cell phone, how many expect to replace their phone within the next 12 months?
37,000,000
(c) If the results of the survey were claimed to indicate that 7.7% of adults in this country who own digital cameras plan to replace their camera in the next 12 months, would you say this statement is descriptive or inferential? Why?
The statement is inferential, because the survey reports on a sample of the country's population. This claim requires an inference to the population.
2.1.11 In a poll, a random sample of 2163 adults (aged 18 and over) was asked, "When you see an ad emphasizing that a product is made in your country, are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey are presented in the side-by-side graph. Complete parts (a) through (d) below.
(a) What proportion of 18- to 34-year-old respondents are more likely to buy when made in their country? What proportion of 35- to 44-year-old respondents are more likely to buy when made in their country?
The proportion of 18- to 34-year-old respondents is 0.36.
The proportion of 35- to 44-year-old respondents is 0.5.
(b) What age group has the greatest proportion who are more likely to buy when made in their country?
55+ yrs
(c) Which age group has a majority of respondents who are less likely to buy when made in their country?
18-34 yrs
(d) What is the apparent association between age and likelihood to buy when made in their country?
As age increases so does likelihood to buy homegrown.
2.1.13 A national survey asked people, "How often do you eat out for dinner, instead of at home?" The frequencies were as follows. Complete parts (a) through (g).
(a) Construct a relative frequency distribution of the data.
(b) What percentage of respondents answered "Always"?
1.2%
(c) What percentage of respondents answered "Never" or "Rarely"?
36.2%
(d) Construct a frequency bar graph. Choose the correct answer below.
(e) Construct a relative frequency bar graph. Choose the correct answer below.
(f) Construct a pie chart. Choose the correct answer below.
(g) Suppose a person claims that, "1.2% of all people in the nation always eat out." Is this a descriptive or inferential statement?
inferential
2.1.18 The following data represent the number of male and female murder victims by age in a recent year. Use the data to complete parts a-d.
(a) Construct a relative frequency distribution for males. Express each relative frequency as a decimal.
(b) Construct a relative frequency distribution for females. Express each relative frequency as a decimal.
(c) Construct a side-by-side relative frequency bar graph. In the bar graphs below, the blue (left-side) bars represent males and the orange (right-side) bars represent females.
(d) Compare each gender's age percentages.
The percentages of male murder victims for each age group are fairly close to the percentages of female murder victims for the same age group.
2.1.25 According to a language association, the number of college students studying a foreign language is increasing. The following data represent the foreign language being studied based on a simple random sample of 30 students learning a foreign language. Complete parts (a) through (e).
Chinese Chinese Spanish German Italian Spanish
Chinese French Spanish German Japanese Spanish
Spanish Spanish Russian Spanish Chinese Russian
Spanish Spanish Latin Italian Spanish Russian
Spanish French Spanish Spanish Russian French
(a) Construct a frequency distribution.
(b) Construct a relative frequency distribution.
(c) Draw a frequency bar graph.
(d) Draw a relative frequency bar graph.
(e) Draw a pie chart.
2.1.27 The data in the accompanying table represent the land area and highest elevation for each of seven states of a country. Complete parts (a) and (b).
(a) Would it make sense to draw a pie chart for land area?
Yes
If it makes sense, draw a pie chart. [Show Less]