Question

when hana goes to the mall, she always buys the same lunch and also buys some books. the table shows the number of books she buys, ( x ), and the total amount of money she spends, ( y ).

( x )	1	2	2	3	4

if this data were displayed in a scatter plot, select all statements that would be true about a good trend line summarizing the data.
a. the slope approximates the average cost of one book.
b. the slope approximates the cost of hana’s lunch.
c. the ( y )-intercept approximates the average cost of one book.
d. the ( y )-intercept approximates the cost of hana’s lunch.
e. the trend line goes through the origin.

Explanation:

Step1: Define linear model

Let the linear model be $y = mx + b$, where $m$ is the slope, $b$ is the y-intercept. Here, $y$ is total spending, $x$ is number of books. The model represents $y = (\text{cost per book})x + \text{lunch cost}$, so $m$ = cost per book, $b$ = lunch cost.

Step2: Calculate slope formula

The slope $m$ is calculated as:
$$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$
First compute sums:
$\sum x = 1+2+2+3+4=12$, $\sum y=17+21+23+26+34=121$
$\sum xy=(1*17)+(2*21)+(2*23)+(3*26)+(4*34)=17+42+46+78+136=319$
$\sum x^2=1^2+2^2+2^2+3^2+4^2=1+4+4+9+16=34$, $n=5$

Step3: Compute slope value

$$m = \frac{5*319 - 12*121}{5*34 - 12^2} = \frac{1595 - 1452}{170 - 144} = \frac{143}{26} \approx 5.5$$
This is the average cost of one book.

Step4: Calculate y-intercept

Use $b = \frac{\sum y - m\sum x}{n}$
$$b = \frac{121 - 5.5*12}{5} = \frac{121 - 66}{5} = \frac{55}{5} = 11$$
This is the cost of lunch (fixed cost when $x=0$, no books bought).

Step5: Evaluate each option

• A: Slope ≈5.5 is average book cost: True
• B: Slope is book cost, not lunch: False
• C: Y-intercept is lunch cost, not book cost: False
• D: Y-intercept=11 is lunch cost: True
• E: Y-intercept=11≠0, so line does not pass through origin: False

Answer:

A. The slope approximates the average cost of one book.
D. The y-intercept approximates the cost of Hana's lunch.