Question

In retail stores, there is a lot of competition for shelf space. There are national brands

For most products, and many stores carry their own line of in-house brands, too.

Because shelf space is not infinite, the question is how many linear feet to allocate to

Each product and which shelf (top, bottom, or somewhere in the middle) to put it on.

The middle shelf is the most popular and lucrative, because many shoppers, if

Undecided, will simply pick the product that is at eye level.

A local store that sells many upscale goods is trying to determine how much shelf space

To allocate to its own brand of men’s personal-grooming products. The middle shelf

Space is randomly varied between $3$and $6$linear feet over the next $12$weeks, and

Weekly sales revenue (in dollars) from the store’s brand of personal-grooming products

For men is recorded. Here is some computer output from the study, along with a

Scatterplot:

a. Describe the relationship between shelf length and sales.

b. Write the equation of the least-squares regression line. Be sure to define any

variables you use.

c. If the store manager were to decide to allocate $5$linear feet of shelf space to the

store’s brand of men’s grooming products, what is the best estimate of the weekly

sales revenue?

d. Interpret the value of s.

e. Identify and interpret the coefficient of determination.

Short answer

(a) Strong positive and linear relationship are the relations between length and sales.

(b) Equation is $\mathrm{y}=317.94+152.68\mathrm{x}$where y is sales and x is length.

(c) Best estimation is $\$1081.34$.

(d) The value of s is $22.9212$.

(e) Coefficient of determination is$98.2\%$.

Step-by-step solution

Part (a) Step 1: Given Information

We need to describe the relation between length and sales.

Part (a) Step 2: Explanation

Direction is positive because patterns in scatterplot slopes upwards and form is linear because There is no strong curvature in scatterplot. Strength is strong because points are not spread that's Why relation is positive strong and linear .

Part (b) Step 1: Given Information

We need to write the equation of least square regression line.

Part (b) Step 2: Explanation

Let x = length , y =sales

And we know that general equation of least square is $\mathrm{y}=\mathrm{ax}+\mathrm{b}$

We are given the values of a and b in graph

$a=152.680,b=317.94$That's why the equation is:

$y=152.680x+317.94$.

Part (c) Step 1: Given Information

We need to find the best sales revenue.

Part (c) Step 2: Explanation

We have a equation $y=152.680x+317.94$and we have to given the value of x =$5$then y is:

$$\begin{gathered} \mathrm{y}=152.680\times 5+317.94 \\ \mathrm{y}=1081.34 \end{gathered}$$

Part (d) Step 1: Given Information

We need to find s.

Part (d) Step 2: Explanation

The standard deviation of residuals (s) is given in diagram is$s=22.9212$.

Part (e) Step 1: Given Information

We need to find the coefficients of determination.

Part (e) Step 2: Explanation

$98.2\%$of variation in amount of sales in dollars can be accounted for by LSLR that is dependent Upon length . That's why coefficients of determination is $98.2\%$