Question

The marketing manager of a large supermarket chain would like to use shelf space to predict the sales of pet food. For a random sample of 12 similar stores, she gathered the following information regarding the shelf space, in feet, devoted to pet food and the weekly sales in hundreds of dollars. Use these data to answer questions 8 through 10.
Store 1 2 3 4 5 6
Shelf Space 5 5 5 10 10 10
Weekly Sales 1.6 2.2 1.4 1.9 2.4 2.6

Store 7 8 9 10 11 12
Shelf Space 15 15 15 20 20 20
Weekly Sales 2.3 2.7 2.8 2.6 2.9 3.1


What is the estimated regression equation?

A. = 2.63 + 0.724x
B. = 1.45 + 0.724x
C. = 1.45 + 0.074x
D. = 2.63 - 0.174x

Answer 1

Answer:

$m=\frac{27.75}{375}=0.074$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{150}{12}=12.5$

$\bar y= \frac{\sum y_i}{n}=\frac{28.5}{12}=2.375$

And we can find the intercept using this:

$b=\bar y -m \bar x=2.375-(0.074*12.5)=1.45$

So the line would be given by:

$y=0.074 x +1.45$

C. = 1.45 + 0.074x

Step-by-step explanation:

The data given is:

x: 5,5,510,10,10, 15,15,15, 20,20,20

y: 1.6,2.2,1.4, 1.9, 2.4,2.6, 2.3,2.7, 2.8,2.6, 2.9, 3.1

For this case we need to calculate the slope with the following formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i = 150$

$\sum_{i=1}^n y_i =28.5$

$\sum_{i=1}^n x^2_i =2250$

$\sum_{i=1}^n y^2_i =70.69$

$\sum_{i=1}^n x_i y_i =384$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=2250-\frac{150^2}{12}=375$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=384-\frac{150*28.5}{12}=27.75$

And the slope would be:

$m=\frac{27.75}{375}=0.074$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{150}{12}=12.5$

$\bar y= \frac{\sum y_i}{n}=\frac{28.5}{12}=2.375$

And we can find the intercept using this:

$b=\bar y -m \bar x=2.375-(0.074*12.5)=1.45$

So the line would be given by:

$y=0.074 x +1.45$

C. = 1.45 + 0.074x