In retail stores, there is a lot of competition for shelf space. Not only are there national brands for most products, but many stores have their own in-house

brands. Since 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,

since 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 three and six 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. Below is some computer output from the study, along with a scatterplot.

(a) Does it appear that the weekly sales revenue is related to the shelf length allocated to the house brand? Justify your answer.

(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 five 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.

(f) The store manager questions the intercept of the regression line: “Am I supposed to believe that this analysis tells me that I can sell these products with no shelf space?” How do you answer her?

The figure is :

A simple random sample (SRS) of size $n$ is made up of $n$ people chosen from the population with an equal chance of being the sample that is actually chosen.

Output is

The pattern in the scatter plot is linear and positive, according to the provided scatter plot. In addition, there are no outliers in the plot and the points do not deviate from the main trend. As a result, the weekly sales revenue may be considered to be proportional to the length of the house given to the brand of house.

The general equation of the regression line is,

$\stackrel{⏜}{y}=a+bx$

The dependent variable (X) is shelf length, and the dependent variable (Y) is sales, according to the query. The constant value is $317.94$, and the coefficient of the variable "Shelf length" is $317.94,$ according to the provided output.

Thus, $a=318.16and152.680$

Putting the values of $a$ and $b$ in the regression line. Thus:

$$\begin{gathered} \overline{y}=a+bx \\ =317.94+152.680x \end{gathered}$$

Thus, the regression line is$\stackrel{⏜}{y}=317.94+152.680x$

The most accurate prediction of weekly sales income is:

$$\begin{gathered} \stackrel{⏜}{y}=317.94+152.680x \\ =317.94+152.68\left(5\right) \\ =1081.34 \end{gathered}$$

Thus, the best estimate is$\$1081.34$

$s$ is $22.9212$ in the output. It demonstrates that the expected sale of the week is likely to be $\$22.9212$ off from the actual or genuine sale on average.

The coefficient of determination reveals how much variance in the response variable is explained by the predictor variable. The coefficient of determination in this case is $98.2\%$ As a result, the least square line may account for $98.2$ percent of the variation in the response variable.

The data range is between 3 and 6 feet, as can be observed. As a result, stating that the shelf's length is 0 would be incorrect. As a result, he should provide the right response.