Lesson 6.1.4
HW:
Day 1: 6-35 to 6-40
Day 2: 6-41 to 6-46
Learning Target: Scholars will find the least squares regression line (LSRL) using their calculators and
understand that it is the line that minimizes the sum of the squares of the residuals.
In previous problems, each team drew a slightly different line of best fit. Today your class will agree on a
single line of best fit for any particular situation.
6-30. The following table shows data for one season of the El Toro professional basketball team. El Toro
team member Antonio Kusoc was inadvertently left off of the list. Antonio Kusoc played for 2103
minutes. We would like to predict how many points he scored in the season.
1. Obtain a Lesson 6.1.4 Resource Page from your teacher or use the 6-30 & 6-33 Student
eTool (Desmos). Draw a line of best fit for the data and then use it to write an equation that
models the relationship between total points in the season and minutes played.
2. Which data point is an outlier for this data? Whose data does that point represent? What is his
residual?
3. Would a player be more proud of a negative or positive residual?
4. Predict how many points Antonio Kusoc made.
6-31. Different people will come up with different models for the relationship between total points and
minutes played in the previous problem. They will also have different estimates for the number of points
for Antonio Kusoc.
 Your Task: Discuss with your team how you can decide which team’s equation models the data the
best. Compare your team’s model to that of another team. Which of the two models is better?
 What do you think about when deciding where to place a line of best fit?
 What makes one line a better model than another line?
 How can you numerically describe how close the prediction made by the model is to a player’s actual
total points?
 Why is thinking about absolute value important in this problem?
6-32. Sometimes there are several different lines of best fit that can be drawn with the same sum of the
absolute values of the residuals. To create a single unique line of best fit, statisticians use the sum of the
squares of the residuals, instead of the absolute value, to make all the residuals positive.
1. What is the sum of the squares of the residuals for your line of best fit?
2. Compare your result to other teams in your class as directed by your teacher. Did any team have a
better model than yours because they had a smaller sum of the squares of the residuals?
6-33. A least squares regression line (LSRL) is a unique line that has the smallest possible value for the
sum of the squares of the residuals.
1. Your teacher will show you how to use your calculator to make a scatterplot or use the 6-30 & 633 Student eTool (Desmos). (Hand-held graphing calculator instructions can also be downloaded
here.) Be sure to use the checksum at the bottom of the table in problem 6-30 to verify that you
entered the data into your calculator accurately.
2. Your teacher will show you how to find the LSRL and graph it on your calculator. Sketch your
scatterplot and LSRL on your paper.
3. Your teacher will show you how to find the residuals for the LSRL using your calculator. What is
the sum of the squares of the residuals of the LSRL the calculator found? Was it less than your
sum of squares?
4. How many points does the LSRL predict for Antonio Kusoc?
5. Interpret the slope and y-intercept of the model in context. Explain why this LSRL model is not
reasonable for players that played less than about 350 minutes.
6-35. Charlie’s friend is visiting from Texas and asks him, “What does a hamburger cost in this
town?” This caused Charlie to wonder because the price of a hamburger seems to be different at every
eatery. Charlie thinks there may be an association between the amount of meat in the patty and the cost of
the hamburger. He collected the following data.
1. Interpret the slope and y-intercept in context. Does the y-intercept make sense in this
situation?
2. What is the residual for the hamburger with the 3 ounce patty? What does it mean in
context?
3. Charlie’s friend says that in his home town he can buy a 1 pound hamburger for $14.70.
Would this be a reasonable price in Charlie’s town? Show how you know.
6-36. An item currently costing $20 is increasing in cost by 5% per year.
4. What is the multiplier?
5. What will the item cost five years from now?
6. Write an equation to find the cost in n years.
6-37. Solve each system of equations using the method of your choice.
y + x = –2
5x – 3y = 22
8.
x=3
7x – 2y = 24
7.
6-38. If
, find each of the following values.
9.
10.
11.
12.
f(–3)
f(–1.5)
f(–2)
x if f(x) = 5
6-39. Determine the slope and intercepts for the line 2x + 3y = 6 and then draw a graph.
6-40. The graph at right represents the temperature of a cup of hot water that is left on a kitchen
table.
13. What does the horizontal line (asymptote) represent?
14. How would the graph be different if the cup of hot water was left outside during winter
instead?
6-41. Fabienne compared annual grocery bills with the other mothers at her church. She discovered a
linear relationship between the total cost (in dollars) of groceries and the number of miles the mother
lived from the downtown church. 6-41 HW eTool (Desmos).
15. Do you think that the association would be positive or negative? Strong or weak?
16. The upper boundary for Fabienne’s prediction was modeled by g = 11.27 – 0.14d where g
is the cost of groceries (in thousands of dollars) and d is the distance from church
(miles). The lower boundary was g = 7.67 – 0.14d. What is the equation of Fabienne's
line of best fit?
17. Interpret the slope of Fabienne’s model in context.
18. Fabienne lives 8 miles from church. Her residual was $510. How much did she spend on
groceries this year?
6-42. Robbie’s class collected the following view tube data in problem 6-1.
19. Use your calculator to make a scatterplot and graph the least squares regression line
(LSRL). Sketch the graph and LSRL on your paper. Remember to put a scale on the xaxis and y-axis of your sketch. Write the equation of the LSRL rounded to four decimal
places.
20. With your calculator, find the residuals as you did in part (c) of problem 6-33. Make a
table with the distance from wall (inches) as the first column, and residual (inches) in the
second column. What is the sum of the squares of the residuals?
6-43. Solve each equation for x.
21. 10 – 2(2x + 1) = 4(x – 2)
22. 5 – (2x – 3) = –8 + 2x
6-44. The cost of a 55 inch HDTV is decreasing 15% per year.
23. What is the multiplier?
24. If the current cost of the HDTV is $1500, what will it cost in four years?
25. If the current cost of the HDTV is $1500, write an equation for the cost in n years.
6-45. For each graph below, what are the domain and range?
26.
27.
28.
29. Which, if any, of the graphs represents a function?
6-46. For each sequence defined recursively, write the first few terms. Then use the terms to write an
explicit equation.
30. a1 = 17
an+1 = an – 3
31. a1 = 20
an+1 =
· an
Lesson 6.1.4

6-30. See below:
1. Answers will vary. Possible answer p = –50 + 0.4t where p is the total points
scored in a season and t is the time played in minutes.
2. Scottie Sordan is an outlier. Using the model from part (a), his residual is 2491 –
1186 = 1305 pts. Some students may also notice the John Bailey is also far from
the bulk of the data. Distinguishing between types of outliers is left to a later
course.
3. A positive residual is better because the player’s actual points are greater than we
would predict from the model.
4. Answers will vary. Possible answer: –50 + 0.4(2103 ) ≈ 791 pts

6-31. Possible explanation: We draw the line of best fit by trying to get as close as we
can to the most points. The best model is the one that is closest to the most points. One
way to measure how close the line is to the points is with the residuals. We could add up
all the residuals to see how close the line is overall. However, we would have to sum the
absolute value of all the residuals. Absolute value is a measure of distance, so we are
measuring the total distance the prediction model is from the actual observed values. The
sum of the absolute values for the possible model in part (a) of problem 6-30 is 1209.2.

6-32. See below:
2. Sum of the squares of the residuals for the model in part (a) of problem 6-30 is
2,086,163.84 points2.

6-33. See below:
2. y = 0.59x – 208.74
3. 1,502,805.87 points2; yes it is much less than 2,086,163.84 points2.
4. 1042 points if students used the exact equation in their calculator, or 1032 points
if they use the LSRL with rounded values.
5. Slope of 0.59 means that an increase of one minute in playing time is predicted to
result in an increase of 0.59 points in the season. A y-intercept of –208.74 means
that if a player did not play any minutes at all, he would expect to earn negative
points: models are a tool for representing data, but they are not perfect. Under 350
minutes, the model predicts a negative number of points.

6-35. See below:
1. The slope means that for every increase of one ounce in the patty size you can
expect to see a price increase of $0.74. The y-intercept would be the cost of the
hamburger with no meat. The y-intercept of $0.23 seems low for the cost of the
bun and other fixings, but is not entirely unreasonable.
2. One would expect to pay 0.253 + 0.735(3) = $2.46 for a hamburger with a 3 oz
patty while the cost of the given 3 oz patty is $3.20, so it has a residual of $3.20 –
$2.46 = $0.74. The 3 oz burger costs $0.74 more that predicted by the LSRL
model.
3. The LSRL model would show the expected cost of a 16 oz burger to be 0.253 +
0.735(16) = $12.01. 16 oz represents an extrapolation of the LSRL model,
however $14.70 is more than $2 overpriced.

6-36. See below:
1. 1.05
2. 20(1.05)5 = $25.52
3. t(n) = 20(1.05)n

6-37. See below:
1. (2, –4)
2. (3, –
)

6-38.
1.
2.
3.
4.

6-39. m = –

6-40. See below:
1. Room temperature. The hot water will approach room temperature but will never
cool more than that.
2. The asymptote would be lower, but still parallel to the x-axis. If the temperature
outside was below zero, the asymptote would be below the x-axis.

6-41. See below:
1. Answers will vary. Students may say negative because “in-town” prices can be
higher than prices in the outskirts, or out-of-town families may grow some of their
food. Students may say positive because transportation costs make out-of-town
prices higher, or out-of-town families eat at restaurants less. The association is
probably pretty weak.
2. The y-intercept is halfway between 11.27 and 7.67, so the equation is g = 9.47 –
0.14d.
3. For each additional mile from church, we expect families to $140 less for
groceries this year.
4. $8860

6-42. See below:
See below:
–1
2
undefined
–1.8
, (3, 0), (0, 2); See graph below.
2. See scatterplot below. y = 1.6568 + 0.1336x. Students need to round to four
decimals because if they round to fewer decimals, the LSRL gets too far away
from the actual points due to the lack of precision.
3.
See table below; sum of the squares is 0.5881in2
Distance from wall
(in)
144
132
120
96
84
72
60
Residuals (in)
–0.198
0.305
–0.391
0.316
0.219
0.123
–0.374

6-43. See below:
1. x = 2
2. x = 4

6-44. See below:
1. 0.85
2. 1500(0.85)4 ≈ $783
3. an = 1500(0.85)n

6-45. See below:
1. D: −2 ≤ x ≤ 2, R: −3 ≤ y ≤ 2
2. D: x = 2, R: all numbers
3. D: x ≥ −2, R: all numbers
4. Only graph (a).

6-46. See below:
1. an = 20 − 3n
2.
an = 40(
)n