Line of Best Fit

Show your work for each problem.

1.
Which description fits a good line of best fit drawn by eye?
1. a)About half of the points are above it and half below, and it follows the trend
2. b)It joins the first and the last data point
3. c)It passes through as many points as possible
4. d)All points are below it
Answer: About half of the points are above it and half below, and it follows the trend
A good line of best fit has about half of the points above it and half below it, and it follows the direction of the data. It does not have to pass through any point.
2.
A line of best fit passes through the points $(0, 4)$ and $(5, 14)$ . What is its slope?
Answer: 2
$m = \frac{14 - 4}{5 - 0} = \frac{10}{5} = 2$ .
3.
A line of best fit passes through $(0, 2)$ and $(4, 10)$ . Find its equation $y = mx + b$ .
Answer: y = 2x + 2
1. What is the change in y, $10 - 2$ ? 8
2. What is the change in x, $4 - 0$ ? 4
3. What is the slope $m$ ? 2
4. What is the intercept $b$ ? 2
4.
Mia's line of best fit is $y = 2x + 3$ , where $x$ is the number of weeks of training and $y$ is the number of push-ups in one minute. What does the slope $2$ mean?
1. a)Each week of training adds about 2 push-ups
2. b)Mia did 2 push-ups at the start
3. c)Mia trained for 2 weeks
4. d)Mia does one more push-up every 2 weeks
Answer: Each week of training adds about 2 push-ups
The slope is the change in $y$ per unit of $x$ : each extra week of training adds about 2 push-ups. The $3$ is the intercept, the starting value.
5.
A line of best fit drawn by eye passes through $(0, 12)$ and $(4, 4)$ . What is its y-intercept $b$ ?
Answer: 12
The intercept is the value of $y$ when $x = 0$ . The point $(0, 12)$ is on the line, so $b = 12$ .
6.
Is the line $y = x + 1$ a good line of best fit for the points $(1, 3)$ , $(2, 2)$ , $(3, 5)$ , $(4, 4)$ , $(5, 7)$ , $(6, 6)$ ? Count the points above and below it.
Answer: 1
1. How many points are above the line? 3
2. How many points are below the line? 3
3. Is the line a good fit? Type 1 for yes or 0 for no. 1
7.
A student draws a line through a scatter plot of 8 points. 7 points are above the line and 1 is below it, spread evenly from left to right. How should the student fix the line?
1. a)Make the line steeper without moving it
2. b)Nothing, the line is fine
3. c)Move the line up
4. d)Move the line down
Answer: Move the line up
With 7 points above and only 1 below, the line is too low. Since the points are spread evenly along it, the direction is fine: move the line up until about 4 points are on each side.
8.
A trend line passes through $(2, 9)$ and $(6, 21)$ . What is its slope?
Answer: 3
$m = \frac{21 - 9}{6 - 2} = \frac{12}{4} = 3$ .
9.
A line of best fit passes through $(10, 80)$ and $(30, 40)$ . Find its equation $y = mx + b$ .
Answer: y = -2x + 100
1. What is the change in y, $40 - 80$ ? -40
2. What is the change in x, $30 - 10$ ? 20
3. What is the slope $m$ ? -2
4. What is $b = 80 - (-2)(10)$ ? 100
10.
The mean point of a data set is $(4, 10)$ . Which of these lines passes through it?
1. a) $y = 2x + 3$
2. b) $y = x + 5$
3. c) $y = 2x + 1$
4. d) $y = 3x - 2$
Answer: $y = 3x - 2$
At $x = 4$ : $2(4) + 1 = 9$ , $4 + 5 = 9$ , $2(4) + 3 = 11$ , $3(4) - 2 = 10$ . Only $y = 3x - 2$ gives $10$ .
11.
A scatter plot has the points $(2, 5)$ , $(4, 9)$ , $(6, 10)$ , $(8, 16)$ . What is $\bar{y}$ , the y-coordinate of the mean point?
Answer: 10
$\bar{y} = \frac{5 + 9 + 10 + 16}{4} = \frac{40}{4} = 10$ . (And $\bar{x} = 5$ , so the mean point is $(5, 10)$ .)
12.
Data: $(1, 3)$ , $(2, 6)$ , $(3, 6)$ , $(4, 9)$ , $(5, 11)$ . A line of best fit with slope $2$ should pass through the mean point. Find its intercept $b$ .
Answer: 1
1. What is $\bar{x}$ ? 3
2. What is $\bar{y}$ ? 7
3. What is $b = \bar{y} - 2\bar{x}$ ? 1
13.
The line of best fit for a phone's battery level $y$ (%) after $x$ hours of use is $y = -12x + 100$ . Interpret it.
Answer: 5
1. By how many percentage points does the battery level change per hour? (Use a negative number for a drop.) -12
2. What battery level does the line give at 0 hours? 100
3. After how many hours does the line reach 40%? 5
14.
For students between 150 cm and 190 cm tall, the line of best fit between height $x$ (cm) and arm span $y$ (cm) is $y = 1.1x - 18$ . What does the intercept $-18$ tell you?
1. a)It has no real meaning: a height of 0 cm is far outside the data
2. b)Everyone's arm span is 18 cm shorter than their height
3. c)A person 0 cm tall has an arm span of -18 cm, a real prediction
4. d)The arm span grows by 18 cm for each cm of height
Answer: It has no real meaning: a height of 0 cm is far outside the data
The intercept is the predicted arm span at height $0$ cm. No student is 0 cm tall and the data only covers 150 to 190 cm, so the intercept has no real meaning. It only places the line correctly.
15.
A line of best fit has slope $-4$ and passes through the mean point $(5, 30)$ . What is its intercept $b$ ?
Answer: 50
$b = 30 - (-4)(5) = 30 + 20 = 50$ , so the line is $y = -4x + 50$ . Check: $-4 \times 5 + 50 = 30$ .
16.
On a scatter plot, a line of best fit is drawn through the points $(1, 7)$ and $(9, 3)$ on the line. Find its slope and intercept, then the value of the line at $x = 5$ .
Answer: 5
1. What is the slope $m = \frac{3 - 7}{9 - 1}$ ? Write it as a decimal. -0.5
2. What is $b = 7 - m \times 1$ ? 7.5
3. What value does the line give at $x = 5$ ? 5
17.
The mean point of a data set is $(6, 20)$ . A student's line of best fit passes through $(0, 5)$ and $(10, 30)$ . Does it pass through the mean point?
Answer: 1
1. What is the slope of the line? 2.5
2. What is the intercept $b$ ? 5
3. What value does the line give at $x = 6$ ? 20
4. Does the line pass through the mean point? Type 1 for yes or 0 for no. 1
18.
A café's line of best fit for price $x$ (in dollars) and cakes sold per day $y$ passes through the mean point $(4, 18)$ and through $(8, 10)$ . Find its equation and the number of cakes it predicts at a price of 6 dollars.
Answer: 14
1. What is the slope $m = \frac{10 - 18}{8 - 4}$ ? -2
2. What is $b = 18 - m \times 4$ ? 26
3. How many cakes does the line predict at $x = 6$ ? 14