Unit 6 Associations In Data — Unit Plan

Title

Assessment

Lesson 1

Organizing Data

Beach Cleaning

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.

Lesson 2

Plotting Data

Right Side Measurements

The table shows measurements of right hand length and right foot length for 5 people.

	right hand length (cm)	right foot length (cm)
person A	19	27
person B	21	30
person C	17	23
person D	18	24
person E	19	26

Draw a scatter plot for the data.
Circle the point in the scatter plot that represents Person D’s measurements.

Show Solution

Lesson 3

What a Point in a Scatter Plot Means

Quarterbacks

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

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.

scatter plot showing monthly advertising vs monthly income

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.

Lesson 4

Fitting a Line to Data

A 1-Foot Foot

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.

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 $(29.1, 10.4)$ .
About 10.4 cm
About 11.1 cm

Lesson 5

Describing Trends in Scatter Plots

This Is One Way to Do It

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.

Lesson 7

Observing More Patterns in Scatter Plots

Make Your Own Scatter Plot

Draw a scatter plot that shows a positive linear association and clustering.
Draw a scatter plot that shows a negative non-linear association and no clustering.

Show Solution

Sample responses:

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 $y=\frac{4}{5}x+3.5$ , and the dashed line has the equation $y=\frac{7}{10}x+3.8$ .

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 $\frac{7}{10}$ . This means that, for every extra inch in diameter for one of these bushes, the height is expected to be $\frac{7}{10}$ 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.

Lesson 9

Looking for Associations

Guitar and Golf

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.

plays an instrument does not play an instrument total

plays a sport 89% 100%

does not play a sport 71% 100%

Show Solution

Sample response:

	plays an instrument	does not play an instrument	total
plays a sport	2	16	18
does not play a sport	5	2	7
total	7	18	25

	plays an instrument	does not play an instrument	total
plays a sport	11%, since $2 \div 18 \approx 0.11$	89%, since $16 \div 18 \approx 0.89$	100%
does not play a sport	71%, since $5 \div 7 \approx 0.71$	29%, since $2 \div 7 \approx 0.29$	100%

Lesson 10

Using Data Displays to Find Associations

Class Preferences

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

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 $\geq 100$	total
fire	22	28	50
water	22	3	25
total	44	31	75

	power $< 100$	power $\geq 100$	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 $\geq 100$ 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.

Lesson 11

Gone in 30 Seconds

No cool-down

Unit 6 Assessment

End-of-Unit Assessment

Problem 1

Noah gathered data at his school among 7th and 8th graders to see if there was an association between grade level and handedness. The table and graph show his data, but the number of right-handed 8th graders is missing.

	left-handed	right-handed
7th grade	11	72
8th grade	24

Noah found there was no evidence of an association between grade level and handedness. Which of these could be the number of right-handed 8th graders?

107

157

Show Solution

157

Problem 2

Here is a scatter plot:

Scatterplot, x, y. Points start at point 2 comma 2 point 1 and trend right and down toward 5 point 5 comma point 2.

The graph of what linear equation is a good fit for this data?

$y=\text-\frac13x+2$

$y=\text-\frac13x+6$

$y = \frac 1 3 x + 2$

$y = \frac 1 3 x + 6$

Show Solution

$y=\text-\frac13x+2$

Problem 3

Select all the relationships that demonstrate a positive association between variables.

Outside temperature and cost to heat a home

Length of time walked and distance traveled

Pounds of cherries bought and amount of money spent on cherries

Speed of a train and the amount of time it takes for the train to get to its destination

Number of people in a grocery check-out line and amount of time waiting in line

Show Solution

B, C, E

Problem 4

Draw a scatter plot that shows a positive, linear association and has 1 clear outlier. Circle the outlier.
Draw a scatter plot that shows a negative association that is not linear.

Show Solution

Answers vary.

Minimal Tier 1 response:

Work is complete and correct.
Sample:

Plot shows points nearly in the same line with a positive slope, and 1 circled point not near the line.
Plot shows points that are not nearly in the same line, with a generally negative trend.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: One plot shows a significant error, but the other is correct; outlier not circled; plots have reversed negative and positive associations, but otherwise all work is correct.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Both plots show errors beyond the ones listed in Tier 2.

Problem 5

Jada surveyed all 7th and 8th graders at her school about whether they have pets. Complete the missing entries in this two-way table.

	has pet	has no pet	total
7th grade	102		150
8th grade		68	175
total

Show Solution

	has pet	has no pet	total
7th grade	102	48	150
8th grade	107	68	175
total	209	116	325

Problem 6

At a school social, children attend with family members. Everyone had a choice between a sweet snack and a salty snack. Here is a two-way table showing the number of adults and children who made each choice of snack.

	sweet snack	salty snack	total
adult	57	88	145
child	77	31	108
total	134	119	253

Complete the table with relative frequency by row. Round to the nearest percent.

sweet snack salty snack total

adult 100%

child 100%
Make a segmented bar graph to represent the data in your table. Use one bar for each row of the table.

Show Solution

sweet snack salty snack total

adult 39% 61% 100%

child 71% 29% 100%

<p>Two bar graphs one for adult and one for child.</p>

Minimal Tier 1 response:

Work is complete and correct.
Sample:

The table is completely correct.
The segmented bar graphs are accurate.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Percentages in table are nearly correct but completely due to a calculation or rounding error; segmented bar graphs are inaccurate.
Acceptable errors: Segmented bar graph is correct based on incorrect percentages.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: A regular bar graph is drawn instead of a segmented bar graph; percentages in table are very inaccurate, notably if outside 0–100% range or if using an incorrect total as the basis for the percentages.

Problem 7

Lin opened a lemonade stand during the summer. She noticed that she sold more lemonade on warmer days. For each day she sold lemonade, she plotted the point $(t,c)$ , where $t$ represents daily high temperature in degrees Fahrenheit and $c$ represents cups of lemonade sold.

Scatterplot, temperature, degrees Fahrenheit, cups sold. Points begin at 70 comma 50 and trend up and right toward 87 comma 97.

On the same axes, draw a line that you think is a good fit for the data.
A computer program found that the line $c = 2t - 89$ is a good fit for the data. Use this equation to predict how many cups of lemonade Lin might sell on a day when the daily high temperature is 74 degrees Fahrenheit.
The daily high temperature this Sunday is expected to be 5 degrees warmer than the daily high temperature this Saturday. Using the line $c = 2t - 89$ , how many more cups of lemonade should Lin expect to sell on Sunday than Saturday? Explain or show your reasoning.

Show Solution

Sample response:
59 cups. $2 \boldcdot 74 - 89 = 59$ .
10 more cups. The slope of the line is 2. This means that for each one-degree increase in temperature, Lin can expect to sell about two more cups of lemonade. If the temperature increases by 5 degrees, she can expect to sell about 10 more cups of lemonade.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

The line reasonably represents a fitting line for the data.
59
10, because the slope is 2 and $2 \boldcdot 5 = 10$ .

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Line of fit is not close but still generally follows direction of data; minor visible calculation error gives incorrect number of cups for 74 degrees; explanation missing for why 10 more cups are expected on Sunday.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Line of fit is missing or very inaccurate; incorrect answer given for part b with no work shown or explanation; incorrect application of rule $c = 2t - 89$ .

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two or more error types from Tier 3 response.