Residuals

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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
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) $y - \hat{y}$
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
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
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)
6.
The regression line is $\hat{y} = 5x + 50$ . A data point is $(2, 58)$ . What is its residual?
7.
A student's observed test score is $80$ and the regression line predicts $74.5$ . What is the residual?
8.
A point has a residual of $-4$ and its predicted value is $\hat{y} = 30$ . What is the observed value $y$ ?
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?
10.
The residuals of a regression line are $1$ , $-3$ , $2$ , $0$ , $-1$ and $1$ . What is the sum of squared residuals?
11.
The regression line is $\hat{y} = 3x + 4$ . Find the residual of the point $(5, 21)$ .
12.
The regression line is $\hat{y} = 2x + 10$ . Find the residual of the point $(6, 19)$ and say where the point lies.
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.
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.
15.
For the points $(1, 4)$ , $(2, 4)$ and $(3, 8)$ and the line $\hat{y} = 2x + 1$ , find the sum of squared residuals.
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?
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 $y$ ?
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.