Question

linear regression worksheet #1
name braylan w date 9/19/25 period 7th

1. a convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the table.

(a) make a scatter plot of the data.
(b) find and graph a linear regression equation that models the data.
equation:
(c) use the model to predict soft - drink sales if the temperature is 95°f.
(d) using the scatterplot, describe the association you see between the two variables. make sure to mention form, direction and strength.
high temperature (°f) number of cans sold
55 340
58 335
64 410
68 460
70 450
75 610
80 735
84 780

Explanation:

Step1: Denote variables

Let $x$ be the high - temperature and $y$ be the number of cans sold.

Step2: Calculate necessary sums

For $n = 8$ data - points:
Calculate $\sum_{i = 1}^{n}x_{i}=55 + 58+64 + 68+70+75+80+84=554$
$\sum_{i = 1}^{n}y_{i}=340 + 335+410+460+450+610+735+780=4120$
$\sum_{i = 1}^{n}x_{i}^{2}=55^{2}+58^{2}+64^{2}+68^{2}+70^{2}+75^{2}+80^{2}+84^{2}=39274$
$\sum_{i = 1}^{n}x_{i}y_{i}=55\times340+58\times335 + 64\times410+68\times460+70\times450+75\times610+80\times735+84\times780=307790$

Step3: Calculate slope $m$

$m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}$
$m=\frac{8\times307790 - 554\times4120}{8\times39274-554^{2}}$
$m=\frac{2462320-2282480}{314192 - 306916}$
$m=\frac{179840}{7276}\approx24.72$

Step4: Calculate y - intercept $b$

$\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{554}{8}=69.25$
$\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{4120}{8}=515$
$b=\bar{y}-m\bar{x}=515-24.72\times69.25$
$b=515 - 24.72\times69.25=515 - 1711.38=- 1196.38$
The linear regression equation is $y = 24.72x-1196.38$

Step5: Predict sales for $x = 95$

When $x = 95$, $y=24.72\times95-1196.38$
$y = 2348.4-1196.38=1152.02\approx1152$

Step6: Analyze scatter - plot association

Form: The points in the scatter - plot appear to follow a linear form.
Direction: As the temperature ($x$) increases, the number of cans sold ($y$) increases, so it is a positive direction.
Strength: The points are relatively close to a straight - line, so the association is strong.

Answer:

(a) (Scatter - plot should be drawn with temperature on x - axis and number of cans sold on y - axis, plotting the 8 data - points)
(b) Equation: $y = 24.72x-1196.38$ (Graph should be a straight - line with the calculated slope and y - intercept)
(c) Approximately 1152 cans
(d) Form: Linear; Direction: Positive; Strength: Strong