Residuals

Show your work for each problem.

1.
A data point lies below the regression line. What is the sign of its residual?
1. a)negative
2. b)it depends on the slope
3. c)zero
4. d)positive
Answer: negative
Below the line, $y < \hat{y}$ , so $e = y - \hat{y}$ is negative.
2.
Which formula gives the residual of a data point?
1. a) $y \cdot \hat{y}$
2. b) $y + \hat{y}$
3. c) $\hat{y} - y$
4. d) $y - \hat{y}$
Answer: $y - \hat{y}$
The residual is always observed minus predicted: $e = y - \hat{y}$ . The order matters, because $\hat{y} - y$ has the opposite sign.
3.
A residual plot shows a clear U-shaped curve. What does this tell you?
1. a)There is no relationship between x and y
2. b)The residuals do not add up to 0
3. c)A curved model would fit better than a line
4. d)The linear model fits well
Answer: A curved model would fit better than a line
A curved pattern in the residuals means the data bends and a straight line misses that bend, so a curved model would fit better. The residuals of the least-squares line still add up to $0$ .
4.
What is true about the sum of the residuals of the least-squares regression line?
1. a)It is different for every data set
2. b)It is always 0
3. c)It equals the sum of squared residuals
4. d)It is always positive
Answer: It is always 0
For the least-squares line the positive and negative residuals always cancel exactly: $\sum e = 0$ , for every data set. That is why a small sum says nothing about the fit.
5.
Four lines were drawn through the same scatter plot. Their sums of squared residuals are listed below. Which one is the least-squares regression line?
1. a)Line B (SSR = 34)
2. b)Line A (SSR = 40)
3. c)Line D (SSR = 61)
4. d)Line C (SSR = 53)
Answer: Line B (SSR = 34)
The least-squares line is the one with the smallest sum of squared residuals. $34$ is the smallest value, so line B is the least-squares line.
6.
The regression line is $\hat{y} = 5x + 50$ . A data point is $(2, 58)$ . What is its residual?
Answer: -2
$\hat{y} = 5(2) + 50 = 60$ , so $e = 58 - 60 = -2$ . The point is 2 units below the line.
7.
A student's observed test score is $80$ and the regression line predicts $74.5$ . What is the residual?
Answer: 5.5
$e = 80 - 74.5 = 5.5$ . The residual is positive, so the student scored $5.5$ points more than predicted.
8.
A point has a residual of $-4$ and its predicted value is $\hat{y} = 30$ . What is the observed value $y$ ?
Answer: 26
$y = \hat{y} + e = 30 + (-4) = 26$ . A negative residual means the observed value is below the prediction.
9.
A least-squares line was fitted to $5$ points. Four of the residuals are $3$ , $-1$ , $2$ and $-5$ . What is the fifth residual?
Answer: 1
$3 + (-1) + 2 + (-5) = -1$ . All five must add up to $0$ , so the fifth residual is $1$ .
10.
The residuals of a regression line are $1$ , $-3$ , $2$ , $0$ , $-1$ and $1$ . What is the sum of squared residuals?
Answer: 16
$1^2 + (-3)^2 + 2^2 + 0^2 + (-1)^2 + 1^2 = 1 + 9 + 4 + 0 + 1 + 1 = 16$ .
11.
The regression line is $\hat{y} = 3x + 4$ . Find the residual of the point $(5, 21)$ .
Answer: 2
1. What is $3 \times 5$ ? 15
2. What is the predicted value $\hat{y}$ ? 19
3. What is the residual $e = y - \hat{y}$ ? 2
12.
The regression line is $\hat{y} = 2x + 10$ . Find the residual of the point $(6, 19)$ and say where the point lies.
Answer: -3
1. What is the predicted value $\hat{y}$ ? 22
2. What is the residual $e = y - \hat{y}$ ? -3
3. Where does the point lie? below
13.
A café models hot chocolate sales with $\hat{y} = 0.5x + 20$ , where $x$ is the number of customers. On a day with $40$ customers it sold $38$ hot chocolates. Find the residual.
Answer: -2
1. What is $0.5 \times 40$ ? 20
2. What is the predicted value $\hat{y}$ ? 40
3. What is the residual $e = y - \hat{y}$ ? -2
14.
The least-squares line for the points $(1, 6)$ , $(2, 7)$ and $(3, 14)$ is $\hat{y} = 4x + 1$ . Find each residual and their sum.
Answer: 0
1. Residual of $(1, 6)$ ? 1
2. Residual of $(2, 7)$ ? -2
3. Residual of $(3, 14)$ ? 1
4. What is the sum of the three residuals? 0
15.
For the points $(1, 4)$ , $(2, 4)$ and $(3, 8)$ and the line $\hat{y} = 2x + 1$ , find the sum of squared residuals.
Answer: 3
1. Residual of $(1, 4)$ ? 1
2. Residual of $(2, 4)$ ? -1
3. Residual of $(3, 8)$ ? 1
4. What is the sum of the squared residuals? 3
16.
For the points $(1, 3)$ , $(2, 5)$ and $(3, 4)$ , compare line A: $\hat{y} = 0.5x + 3$ and line B: $\hat{y} = x + 2$ . Which one is the least-squares line?
Answer: A
1. Line A: what is the residual of $(2, 5)$ ? 1
2. Line A: what is the sum of squared residuals? 1.5
3. Line B: what is the sum of squared residuals? 2
4. Which line is the least-squares line? A
17.
The least-squares line for four points is $\hat{y} = 2x + 5$ . The residuals of the first three points are $2$ , $-1$ and $-3$ . The fourth point has $x = 8$ . What is its observed value $y$ ?
Answer: 23
1. What is the sum of the first three residuals? -2
2. What must the fourth residual be? 2
3. What is $\hat{y}$ for $x = 8$ ? 21
4. What is the observed value $y$ ? 23
18.
An app's downloads (in thousands) in months $1$ to $6$ were $20, 24, 31, 41, 54, 70$ . The least-squares line is $\hat{y} = 10x + 5$ . Check the residuals to see whether a straight line fits.
Answer: 5, -1, -4, -4, -1, 5
1. Residual for month $1$ ? 5
2. Residual for month $3$ ? -4
3. Residual for month $6$ ? 5
4. The six residuals are $5, -1, -4, -4, -1, 5$ . What pattern do they form in the residual plot? U-shaped curve