Making Predictions from Data

Show your work for each problem.

1.
A regression line is $\hat{y} = 4x + 10$ . What is the predicted value when $x = 5$ ?
1. a)14
2. b)40
3. c)30
4. d)20
Answer: 30
$\hat{y} = 4(5) + 10 = 20 + 10 = 30$ . The answer 20 forgets the intercept, and 14 adds the slope to the intercept ( $4 + 10$ ) and forgets $x$ .
2.
The regression line for a data set is $\hat{y} = 3x + 12$ . Predict $\hat{y}$ for $x = 6$ .
Answer: 30
$\hat{y} = 3(6) + 12 = 18 + 12 = 30$ .
3.
The data in a scatter plot has $x$ -values from $2$ to $10$ . For which $x$ -value is a prediction an interpolation?
1. a)12
2. b)7
3. c)15
4. d)0
Answer: 7
Only $7$ lies inside the range: $2 \le 7 \le 10$ . The values $0$ , $12$ and $15$ are outside it, so predictions there are extrapolations.
4.
A plant's height in mm follows $\hat{y} = 2.5x + 40$ , where $x$ is the number of days (data from day 1 to day 10). Predict the height on day $4$ .
Answer: 50
$\hat{y} = 2.5(4) + 40 = 10 + 40 = 50$ mm. Day 4 is inside the data, so this is interpolation.
5.
A shop models its daily umbrella sales with $\hat{y} = 6x + 20$ , where $x$ is the rainfall in mm (data from 0 to 12 mm). Predict the sales on a day with $7$ mm of rain.
Answer: 62
1. Multiply the slope by x: what is $6 \times 7$ ? 42
2. Add the intercept 20. What is the predicted number of umbrellas? 62
6.
Four data sets each have a regression line. Which correlation coefficient gives the most reliable predictions?
1. a)r = -0.95
2. b)r = 0.1
3. c)r = 0.25
4. d)r = 0.5
Answer: r = -0.95
$|-0.95| = 0.95$ is the closest to $1$ , so the points lie very close to the line. The negative sign only means the line goes down.
7.
A model predicts a test score with $\hat{y} = 4x + 18$ , where $x$ is hours of practice (data from 2 to 15 hours). For which $x$ is the predicted score $50$ ?
Answer: 8
1. Write $50 = 4x + 18$ and subtract 18 from both sides. What is $50 - 18$ ? 32
2. Divide by the slope: what is $32 \div 4$ ? 8
8.
Using $\hat{y} = 5x + 52$ ( $x$ = hours studied, $\hat{y}$ = test score), how many hours of study predict a score of $87$ ?
Answer: 7
$87 - 52 = 35$ and $35 \div 5 = 7$ , so $x = 7$ hours.
9.
A school fair models visitors with $\hat{y} = 14.6x + 23$ , where $x$ is the number of posters (data from 2 to 10 posters). Predict the number of visitors for $6$ posters, rounded to a whole number.
Answer: 111
1. Calculate $14.6 \times 6$ 87.6
2. Add the intercept 23. What is $\hat{y}$ ? 110.6
3. Round to a whole number of visitors 111
10.
The line $\hat{y} = 5x + 52$ was fitted to data from 1 to 8 hours of study. For 12 hours it predicts a score of $112$ out of $100$ . What is the problem?
1. a)The intercept should be 0
2. b)12 hours is outside the data, and a score above 100 is impossible
3. c)Nothing, the prediction is reliable
4. d)The slope should be negative
Answer: 12 hours is outside the data, and a score above 100 is impossible
The calculation $5(12) + 52 = 112$ is correct, but it is an extrapolation: the pattern was only seen from 1 to 8 hours, and scores stop at 100.
11.
While gaming, a phone's battery level follows $\hat{y} = 100 - 12x$ (percent), where $x$ is hours of gaming (data from 0 to 6 hours). Predict the battery level after $4.5$ hours.
Answer: 46
1. Calculate $12 \times 4.5$ 54
2. Subtract from 100: what is the predicted battery level in percent? 46
12.
With the battery model $\hat{y} = 100 - 12x$ ( $x$ in hours, $\hat{y}$ in percent), after how many hours does the model predict a battery level of $40\%$ ?
Answer: 5
$40 = 100 - 12x \Rightarrow 12x = 60 \Rightarrow x = 5$ hours, which is inside the data from 0 to 6 hours.
13.
A regression model says Tom needs $5.6$ hours of study to reach his target score. Which plan makes the most sense?
1. a)Study 5 hours
2. b)Study about 6 hours
3. c)Study exactly 5.6 hours, not a minute more
4. d)Study 0.6 hours
Answer: Study about 6 hours
To reach a target you round up: about 6 hours. Rounding down to 5 hours predicts too low a score, and the exact 5.6 hours pretends the estimate is perfectly precise.
14.
A regression line has slope $m = 2.5$ and passes through the point of means $(\bar{x}, \bar{y}) = (4, 30)$ . Predict $\hat{y}$ for $x = 6$ .
Answer: 35
1. Find the intercept: $b = \bar{y} - m\bar{x} = 30 - 2.5 \times 4$ 20
2. Calculate $2.5 \times 6$ 15
3. Add the intercept: what is $\hat{y}$ ? 35
15.
A growth model for children aged 2 to 10 is $\hat{y} = 6x + 77$ (height in cm, age $x$ in years). A student uses it to predict the height of a 40-year-old.
Answer: 317
1. Calculate $6 \times 40$ 240
2. Add 77. What height in cm does the model predict? 317
3. Is 317 cm a realistic height for an adult? no
16.
A regression line is $\hat{y} = 2.4x + 15.6$ . For which $x$ is the predicted value $39.6$ ?
Answer: 10
$39.6 = 2.4x + 15.6 \Rightarrow 24 = 2.4x \Rightarrow x = 10$ . Check: $2.4(10) + 15.6 = 39.6$ .
17.
A class has two models for the test score: hours studied, $\hat{y} = 5x + 52$ with $r \approx 0.99$ , and hours of sleep, $\hat{y} = 2x + 59$ with $r \approx 0.21$ . Both are used for $x = 6$ hours.
Answer: 82
1. Predict with the study model for $x = 6$ 82
2. Predict with the sleep model for $x = 6$ 71
3. Which prediction is more reliable? the study model
18.
With $\hat{y} = 5x + 52$ ( $x$ = hours studied, data from 1 to 8 hours), Mia wants a score of at least $90$ . How many whole hours should she plan?
Answer: 8
1. Write $90 = 5x + 52$ . What is $90 - 52$ ? 38
2. Divide by the slope: what is $38 \div 5$ ? 7.6
3. How many whole hours does she need to reach at least 90? 8