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Question 1

Skill question

Advertising budget ( $\$ k) and sales revenue ( $\$ k): $(2,30)$ , $(4,45)$ , $(6,55)$ , $(8,70)$ , $(10,85)$ , $(12,95)$ . Determine the regression line, forecast revenue at $15$ k, and comment on reliability.

Question 2

Given the following bivariate data: $(1, 2), (2, 3), (3, 5), (4, 4), (5, 6)$

(a) Find the equation of the least-squares regression line of $y$ on $x$ .

(b) Hence, estimate the value of $y$ when $x = 6$ .

(c) Comment on the reliability of your estimate in part (b).

[10]

Question 3

Given the regression equation $\hat{y} = 2.5x - 1.3$ fitted to data with $x$ in the range $[1, 5]$ , calculate the predicted value of $y$ when $x = 10$ and comment on the reliability of this prediction.

[4]

Question 4

The following data shows the relationship between the outside temperature, $x$ °C, and the number of ice-cream units sold, $y$ , over seven days:

$(20, 200), (22, 215), (24, 230), (26, 250), (28, 270), (30, 290), (32, 310)$

1.

(a) Find the equation of the regression line of $y$ on $x$ .

[4]

2.

(b) Estimate the number of ice-cream units sold on a day when the temperature is $35$ °C.

[2]

3.

(c) Comment on the reliability of the estimate found in part (b).

[2]

Question 5

Given three data points $(10, 5)$ , $(15, 9)$ , and $(20, 14)$ :

(a) Find the least-squares regression line of $y$ on $x$ .

(b) Use your equation to forecast the value of $y$ when $x = 25$ .

(c) Comment on the reliability of this forecast.

[11]

Question 6

The following table shows the pressure $x$ , in atm, and temperature $y$ , in K, for a sample of gas.

$\begin{array}{|l|c|c|c|c|c|} \hline x \text{ (atm)} & 1 & 2 & 3 & 4 & 5 \\ \hline y \text{ (K)} & 200 & 250 & 300 & 350 & 400 \\ \hline \end{array}$

(a) Find the equation of the regression line of $y$ on $x$ .

(b) (i) Forecast the temperature at a pressure of $6\text{ atm}$ .

(ii) Comment on the reliability of this forecast.

[10]

Question 7

The following table shows the age, $x$ years, and the systolic blood pressure, $y$ mmHg, of six individuals.

\begin{array}{|l|c|c|c|c|c|c|} \hline \text{Age } (x) & 30 & 40 & 50 & 60 & 70 & 80 \\ \hline \text{Blood pressure } (y) & 120 & 130 & 140 & 150 & 160 & 170 \\ \hline \end{array}

1.

Find the equation of the regression line of $y$ on $x$ in the form $y = mx + c$ .

[4]

2.

Use your regression line to estimate the systolic blood pressure of an individual aged 85.

[2]

3.

Comment on the reliability of the estimate found in part 2.

[2]

Question 8

The following data shows the time in hours ( $x$ ) and the number of units produced ( $y$ ) for a manufacturing process:

$(1,15)$ , $(2,22)$ , $(3,28)$ , $(4,34)$ , $(5,38)$ , $(6,45)$

(a) Determine the regression line of $y$ on $x$ .

(b) Predict the production at $x = 8$ hours.

(c) Comment on the reliability of this prediction.

[8]

Question 9

The following table shows the distance travelled in miles, $x$ , and the corresponding fuel consumption in gallons, $y$ , for a particular vehicle.

Distance ( $x$ )	50	100	150	200	250	300	350	400
Fuel ( $y$ )	2.8	5.6	8.3	11.1	13.9	16.7	19.4	22.2

(a) Find the equation of the regression line of $y$ on $x$ .

(b) Use your equation to estimate the fuel consumption for a distance of 450 miles.

(c) Comment on the reliability of this estimate.

[8]

Question 10

The following table shows the number of years of experience, $x$ , and the annual salary, $y$ (in thousands of dollars), for eight employees at a certain company.

Years of experience ( $x$ )	1	3	5	7	9	11	13	15
Salary ( $y$ )	30	38	46	55	63	70	78	85

(a) Determine the equation of the regression line of $y$ on $x$ .

(b) Use your equation to forecast the annual salary of an employee with 12 years of experience.

(c) Comment on the reliability of this forecast.

[8]

Question 11

The number of hours spent studying, $x$ , and the corresponding exam score, $y$ , for five students are recorded in the following table.

\begin{array}{|l|c|c|c|c|c|} \hline \text{Hours studied } (x) & 2 & 4 & 6 & 8 & 10 \\ \hline \text{Exam score } (y) & 65 & 75 & 83 & 90 & 95 \\ \hline \end{array}

1.

Find the equation of the least squares regression line of $y$ on $x$ .

[4]

2.

Use your equation to estimate the exam score for a student who studied for $9$ hours.

[2]

3.

Comment on the reliability of your estimate in part (b).

[1]

Question Type 3: Forecasting another value given a linear fit Exercises