Question

The marketing manager of a large supermarket chain faced the business problem of determining the effect on the sales of pet food of shelf space and whether the product was placed at the front ​(equals​1) or back ​(equals ​0) of the aisle. Data are collected from a random sample of 12​ equal-sized stores and are given below. Complete parts​ (a) through​ (g). For parts​ (a) through​ (d), do not include an interaction term.

Store	Shelf_Space_(Feet)	Location	Weekly_Sales_($)
1	5	Back	160
2	5	front	230
3	5	back	140
4	10	back	200
5	10	front	290
6	10	back	210
7	15	Back	230
8	15	Back	280
9	15	Front	270
10	20	Back	250
11	20	Back	280
12	20	Front	300

a. State the multiple regression equation that predicts weekly sales based on shelf​ space,

Upper X 1​, and​ location, Upper X 2.

ModifyingAbove Upper Y with caret Subscript iequalsplus​( ​)Upper X Subscript 1 iplus​()Upper X Subscript 2 i

​(Round to two decimal places as needed. Do not include the​ $ symbol in your​ answers.)

b. Interpret the regression coefficients in​ (a).

Holding constant whether the product is at the front or back of​ aisle, for each increase of 1 foot of shelf​ space, the predicted sales is estimated to increase by ​$___

.

Holding constant the amount of shelf​ space, the presence of the product at the [front , back ]

of the aisle is estimated to increase the sales by ​$ over the sales of a similar item at the [front , back ] of the aisle. ​(Round to two decimal places as​ needed.)

c. At the 0.05 level of​ significance, determine whether each independent variable makes a contribution to the regression model.

Test the first independent​ variable, Shelf Space.

Determine the null and alternative hypotheses.

Upper H 0​: beta 1 [not equals , greater than or equals ,less than or equals ,equals ] 0

Upper H 1​: beta 1 [greater than , equals ,not equals ,less than ] 0

The test statistic for the first independent​ variable, Shelf Space​, is t Subscript STAT

equals. ​(Round to three decimal places as​ needed.)

The​ p-value for the first independent​ variable, Shelf Space, is .

​(Round to four decimal places as​ needed.)

Since the​ p-value is [less , greater ] than the value of alpha​, [ do not reject ,reject ] the null hypothesis. The first independent​ variable, Shelf Space ​, [ does not appear , appears ] to make a contribution to the regression model.

Test the second independent​ variable, Location.

Determine the null and alternative hypotheses.

Upper H 0​: beta 2 [greater than or equals , not equals ,equals , less than or equals ] 0

Upper H 1​: beta 2 [less than , greater than , equals , not equals ] 0

The test statistic for the second independent​ variable, Location​, is t Subscript STAT

equals. ​(Round to three decimal places as​ needed.)

The​ p-value for the second independent​ variable, Location, is .​(Round to four decimal places as​ needed.)

Since the​ p-value is [greater , less ] than the value of alpha ​, [do not reject , reject ] the null hypothesis. The second independent​ variable, Location​, [ does not appear , appears ] to make a contribution to the regression model.

d. Construct and interpret a​ 95% confidence interval estimate of the population slope of the relationship between weekly sales and the amount of shelf space.

Taking into account the effect of [Shelf Space, Location, Weekly Sales, ] the estimated effect of a 1 foot increase in the amount of shelf space is to change the [Weekly Sales ,Location ,Shelf Space ] by ​$ to ​$ . ​(Round to two decimal places as needed. Use ascending​ order.)

e. Construct and interpret a​ 95% confidence interval estimate of the population slope of the relationship between weekly sales and the location of the product.

Taking into account the effect of [Shelf Space, Weekly Sales, Location, ] the estimated effect of being at the front of the aisle instead of the back of the aisle is to change the [ Location , Weekly Sales ,Shelf Space ] by ​$ to ​$. ​(Round to two decimal places as needed. Use ascending​ order.)

Answer 1

a. State the multiple regression equation that predicts weekly sales based on shelf​ space,

Upper X 1​, and​ location, Upper X 2.

ModifyingAbove Upper Y with caret Subscript iequalsplus​( ​)Upper X Subscript 1 iplus​()Upper X Subscript 2 i

the regression line is

weekly=137.083 + 6.533 *shelf+ 53.750 * location (front)

b. Interpret the regression coefficients in​ (a).

Holding constant whether the product is at the front or back of​ aisle, for each increase of 1 foot of shelf​ space, the predicted sales is estimated to increase by 6.533

.

Holding constant the amount of shelf​ space, the presence of the product at the [front , back ]

of the aisle is estimated to increase the sales by ​$ over the sales of a similar item at the [front , back ] of the aisle. ​(Round to two decimal places as​ needed.)

so keeping shelf as constant if we get location=front then weekly will increase by 53.75

if location = back then weekly will increase by 0

c. At the 0.05 level of​ significance, determine whether each independent variable makes a contribution to the regression model.

so can note that the p-value corresponding shelf and location is 0.000652 and 0.006387 are both less than 0.05

so we can say both coefficients are significantly effective

Test the first independent​ variable, Shelf Space.

Determine the null and alternative hypotheses.

Upper H 0​: beta 1 = 0

Upper H 1​: beta 1 not equals 0

The test statistic for the first independent​ variable, Shelf Space​, is t Subscript STAT

equals. ​(Round to three decimal places as​ needed.) 5.092

The​ p-value for the first independent​ variable, Shelf Space, is .0.0007

​(Round to four decimal places as​ needed.)

Since the​ p-value is [less ,] than the value of alpha​, [reject ] the null hypothesis. The first independent​ variable, Shelf Space ​, [ , appears ] to make a contribution to the regression model.

Test the second independent​ variable, Location.

Determine the null and alternative hypotheses.

Upper H 0​: beta 2 [equals ] 0

Upper H 1​: beta 2 [not equals ] 0

The test statistic for the second independent​ variable, Location​, is t Subscript STAT

equals. ​(Round to three decimal places as​ needed.) 3.533

The​ p-value for the second independent​ variable, Location, is .​(Round to four decimal places as​ needed.) 0.0064

Since the​ p-value is [ less ] than the value of alpha ​, [reject ] the null hypothesis. The second independent​ variable, Location​, [ appears ] to make a contribution to the regression model.

d. Construct and interpret a​ 95% confidence interval estimate of the population slope of the relationship between weekly sales and the amount of shelf space. [3.63,9.44]

Taking into account the effect of [Shelf Space, Location, Weekly Sales, ] the estimated effect of a 1 foot increase in the amount of shelf space is to change the [Weekly Sales ,Location ,Shelf Space ] by ​$ to ​$ . ​(Round to two decimal places as needed. Use ascending​ order.)

e. Construct and interpret a​ 95% confidence interval estimate of the population slope of the relationship between weekly sales and the location of the product. [19.33, 88.17]

Taking into account the effect of [Shelf Space, Weekly Sales, Location, ] the estimated effect of being at the front of the aisle instead of the back of the aisle is to change the [ Location , Weekly Sales ,Shelf Space ] by ​$ to ​$. ​(Round to two decimal places as needed. Use ascending​ order.)