20 volunteers are cleaning the litter from a beach. The number of minutes each volunteer has worked and the number of meters left to clean on their section are recorded.
Here is a scatter plot that shows the data for each volunteer.
Label the vertical axis of the scatter plot.
If a volunteer has worked 45 minutes, should they have closer to 60 meters or 120 meters of beach left to clean? Explain your reasoning.
Show Solution
Sample response: beach left to clean (meters)
60 meters. Sample reasoning: When the time spent cleaning increases, the amount of beach left to clean tends to decrease. To keep in line with the rest of the data, the length left to clean should be closer to 60 meters than 120 meters.
In football, a quarterback can be rated by a formula that assigns a number to how well they play.
A higher number generally means they played better.
Here are a table and scatter plot that show ratings and wins for quarterbacks who started every game in a season.
player
quarterback rating
number of wins
A
93.8
4
B
102.2
12
C
93.6
6
D
89
8
E
88.2
5
F
97
7
G
88.7
6
H
91.1
7
I
92.7
10
J
88
10
K
101.6
9
L
104.6
13
M
84.2
6
N
99.4
15
O
110.1
10
P
95.4
11
Q
88.7
11
A scatterplot. The horizontal axis is labeled “quarterback rating” and the numbers 80 through 120, in increments of 10, are indicated. The vertical axis is labeled “number of wins” and the numbers 0 through 20, in increments of 5, are indicated. The data are as follows: 84 point 2 comma 6. 88 comma 10. 88 point 2 comma 5. 88 point 7 comma 6. 88 point 7 comma 11. 89 comma 8. 91 point 1 comma 7. 92 point 7 comma 10. 93 point 6 comma 6. 93 point 8 comma 4. 95 point 4 comma 11. 97 comma 7. 99 point 4 comma 15. 101 point 6 comma 9. 102 point 2 comma 12. 104 point 6 comma 13. 110 point 1 comma 10.
Circle the point in the scatter plot that represents Player K’s data.
Which quarterback’s data are represented by the point farthest to the left?
Player R is not included in the table. He has a quarterback rating of 99.4 and his team won 8 games. On the scatter plot, plot a point that represents Player R’s data.
Show Solution
The circled point on the scatter plot
Player M
The added point to the scatter plot, plotted larger for visibility
Section A Check
Section A Checkpoint
Problem 1
A business keeps track of the amount it spends on advertising each month and the amount of income it makes that month. The first 10 months have already been plotted in the scatter plot.
The point that represents June is at (26,76.5). What does this point mean for the business?
In November, the business spent $25,000 on advertising and had an income of $95,000. In December, the business spent $30,000 on advertising and had an income of $105,000. Add these points to the scatter plot.
After looking at this data, would you suggest the business spend more on advertising in January or not? Explain your reasoning.
Show Solution
Sample response: It means that the business spent $26,000 on advertising and had $76,500 in income during June.
Additional dots plotted at (25,95) and (30,105).
Sample response: I would suggest the business spend more on advertising. The data show that as more is spent on advertising, more income is made.
Here is a scatter plot that shows lengths and widths of 20 left feet, together with the graph of a model of the relationship between foot length and width.
A scatterplot. Horizontal, from 20 to 32, by 2's, labeled foot length in centimeters. Vertical, from 7 to 12, by 1’s, labeled foot width in centimeters. 20 dots trend upward and to the right. Line drawn, trends linearly upward and right with 11 dots above lie and 9 below. No dots lie on the line. The line begins at about point 21 point 9 comma 9 and ends at about 31 point 25 comma 11 point 5.
Draw a box around the point that represents the foot with length closest to 29 cm.
What is the approximate width of this foot?
What width does the model predict for a foot with length 29 cm?
Show Solution
A box is drawn around the point at approximately (29.1,10.4).
Elena said, “I think this line is a good fit because half of the points are on one side of the line and half of the points are on the other side.” Do you agree? Explain your reasoning.
A scatterplot. Horizontal, from 0 to 12, by 2’s. Vertical, from 0 to 80, by 20’s. Data trends downward and to right. Line of best fit drawn, goes slightly upward and to right. 10 data points above and below line.
Noah said, “I think this line is a good fit because it passes through the leftmost point and the rightmost point.” Do you agree? Explain your reasoning.
A scatterplot. Horizontal, from 0 to 12, by 2’s. Vertical, from 0 to 80, by 20’s. Data trends downward and to right. Line of best fit drawn, goes downward and to right. 14 data points below line, 4 points above line, and two points on line.
Show Solution
Disagree. Sample response: The line is not a good fit because the data show a negative association, but the line has a positive slope.
Disagree. Sample responses: The line is not a good fit because most of the points are below it and the trend of the scatter plot is steeper than the slope of the graph.
Here is a scatter plot that shows the years when some used cars were made and their prices in 2016 together with the graph of a linear model for the relationship between year and price in dollars.
Is the slope positive or negative?
Which of these values is closest to the slope of the linear model shown in the scatter plot?
1,000
3,000
-1,000
-3,000
Use the value you selected to describe the meaning of the slope in this context.
Show Solution
The slope is positive, because as the year of the car increases, the price tends to increase.
1,000
The model predicts that when a car is made 1 year later, the price is 1,000 dollars higher.
Draw a line on the scatter plot that fits the data well.
Scatterplot. Horizontal from 0 to 10, by 2’s. Vertical from negative 0 to 10, by 2’s. 8 dots clustered in upper left side of graph spread horizontally between 0 and 2 point 5 and vertically from 7 to about 8. 11 dots spread horizontally between 6 and 10 and vertically from 3 to 6.
A new point will be added to the scatter plot with x=4. What do you predict for the y-value of this point if it follows the association of the data already in the scatter plot?
A new point will be added to the scatter plot with x=10. What is an example of a y-value of this point if it is considered an outlier?
Show Solution
Sample responses:
6
10
Section B Check
Section B Checkpoint
Problem 1
The scatter plot shows the height and diameter of a kind of bush that grows naturally in an area with 2 possible linear models.
The solid line has the equation y=54x+3.5, and the dashed line has the equation y=107x+3.8.
Which linear model fits the data better? Explain your reasoning.
What is the slope of the model you chose and what does it mean in this situation?
Does this data show a positive, negative, or no association? Explain your reasoning.
Add a point to the graph that would be considered an outlier.
Show Solution
The dashed line fits better. Sample reasoning: It goes through the middle of the data and follows the trend better than the solid line.
The slope is 107. This means that, for every extra inch in diameter for one of these bushes, the height is expected to be 107 of an inch taller.
It shows a positive association because as the diameter increases, the height tends to also increase.
Any point far away from the other points in the scatter plot.
Problem 2
Do these data show a linear or non-linear association?
Explain your reasoning.
Circle any clusters that appear to be present in the data.
Show Solution
The data show a non-linear association. Sample reasoning: The points do not follow a steadily increasing or decreasing trend. They appear to go up and down as the x-coordinate increases.
A cluster is present for points with x-values between 5 and 20 and another cluster for points with x-values between 30 and 45.
In a class of 25 students, some students play a sport, some play a musical instrument, some do both, and some do neither. Complete the two-way table to show the data from the bar graph.
plays an instrument
does not play an instrument
total
plays a sport
16
does not play a sport
5
total
25
Using the entries from the actual frequency table, complete this table so that it shows relative frequencies based on the rows. Round entries to the nearest percentage point.
Here are a two-way table and segmented bar graph for data from students in 2 classes.
Do they show evidence of differences between the 2 classes?
prefers math
prefers science
prefers recess
class A
6
3
8
class B
8
7
15
Stacked bar graph in yellow, red and blue. Horizontal labeled Class A and Class B. Vertical labeled 0 to 100, by 25's. Yellow represents recess. Red represents science. Blue represents math.
Show Solution
There is no evidence of different preferences associated with each class because the segments in the bars are about the same size.
Section C Check
Section C Checkpoint
Problem 1
In a game, creatures are given an element and a power level. Creatures with power level over 100 have a second skill they can use, so it is useful to separate them by that value.
75 creatures are categorized in the two-way table.
power <100
power ≥100
total
fire
22
28
50
water
22
3
25
total
44
31
75
power <100
power ≥100
total
fire
44%
56%
100%
water
100%
Complete the table with the relative frequency for the water creatures.
Based on the relative frequencies, do you think there is an association between creature element and power? Explain your reasoning.
Show Solution
power <100
power ≥100
total
fire
44%
56%
100%
water
88%
12%
100%
There is an association between creature element and power. Sample reasoning: Because the percentages for the power levels are very different based on creature element, the 2 variables are associated.
