Regression Word Problems

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Show your work for each problem.

1.
The model $\hat{y} = 6x + 48$ predicts a test score $\hat{y}$ (points) from weekly study hours $x$ . What does the slope $6$ mean?
1. a)A student who studies 0 hours scores 6 points
2. b)Each extra hour of study per week goes with about 6 more points, on average
3. c)6 students took part in the study
4. d)The score is always 6 times the number of hours
2.
Using $\hat{y} = 6x + 48$ (test score from weekly study hours $x$ ), predict the score of a student who studies $4$ hours per week.
3.
A regression line was fitted to data with $x$ values from $2$ to $10$ . For which value of $x$ is a prediction an extrapolation?
1. a)5
2. b)15
3. c)7.5
4. d)9
4.
A data point has actual value $y = 80$ , and the regression line predicts $\hat{y} = 74$ for it. What is its residual?
5.
A regression of exam score on study time has $R^2 = 0.64$ . Which statement is correct?
1. a)64% of the data points lie exactly on the line
2. b)64% of the variation in exam scores is explained by the linear relationship with study time
3. c)The correlation coefficient is r = 0.64
4. d)Each extra hour of study adds 0.64 points
6.
The correlation between daily steps and calories burned is $r = 0.7$ . What is $R^2$ ? Give it as a decimal.
7.
A dealer's model for used cars is $\hat{y} = 18000 - 1500x$ , where $\hat{y}$ is the price in dollars and $x$ the age in years (data for cars from 0 to 8 years old). What does the intercept $18000$ mean?
1. a)A new car (age 0) is predicted to cost about 18000 dollars
2. b)A new car is worth 1500 dollars
3. c)After 18000 years the car is worth nothing
4. d)The car loses 18000 dollars each year
8.
A café uses $\hat{y} = 110 - 3x$ to predict the number of hot drinks sold from the morning temperature $x$ in °C (data from $0$ to $20$ °C). How many hot drinks does it predict for $12$ °C?
9.
In a model of delivery times, one delivery has a residual of $-5$ minutes. What does this mean?
1. a)The delivery took 5 minutes longer than predicted
2. b)The model predicted a time of 5 minutes
3. c)The slope of the model is -5
4. d)The delivery took 5 minutes less than predicted
10.
A plant's height is modelled by $\hat{y} = 2.5x + 4$ (cm, $x$ in weeks). By how many cm does the predicted height increase over $4$ more weeks?
11.
The café model is $\hat{y} = 110 - 3x$ (hot drinks, $x$ in °C). On a $15$ °C morning the café sold $75$ hot drinks. Find the residual of this morning.
12.
For the study-hours data, the sum of squared residuals is $SS_{\text{res}} = 40$ and the total variation of the scores is $SS_{\text{tot}} = 400$ . Find $R^2$ and say what percentage of the variation in scores the line explains.
13.
The model $\hat{y} = 6x + 48$ was fitted to students who study from $1$ to $5$ hours per week (tests out of $100$ points). Predict the score for $12$ hours and decide what kind of prediction this is.
14.
The dealer's model is $\hat{y} = 18000 - 1500x$ (price in dollars, $x$ = age in years, data from 0 to 8 years). A $4$ -year-old car was sold for 13000 dollars. Find the predicted price and the residual. Was the car sold above or below the predicted price?
15.
A student counts the push-ups $y$ she can do after $x$ weeks of training:
$x$ 1 2 3 4
$y$ 3 5 6 10
Find the least-squares regression line $\hat{y} = mx + b$ .
16.
The café model is $\hat{y} = 110 - 3x$ (hot drinks, $x$ in °C, data from $0$ to $20$ °C), with $R^2 = 0.9$ . On an $8$ °C morning the café sold $90$ hot drinks. Find the prediction, the residual, the correlation coefficient $r$ and the percentage of variation explained.
17.
An ice-cream stand collected data for temperatures from $10$ to $30$ °C. The regression line of ice creams sold per hour on temperature has slope $1.5$ , and the means are $\bar{x} = 20$ °C and $\bar{y} = 50$ ice creams. Find the intercept, predict sales at $30$ °C, and find the residual of an hour at $30$ °C with $61$ ice creams sold.
18.
With $\hat{y} = 6x + 48$ (test score from weekly study hours, data from $1$ to $5$ hours), how many hours of study does the model say are needed for a predicted score of $84$ ? Can this answer be trusted?