Question

Data from a small bookstore are shown in the accompanying table. The manager wants to predict Sales from Number of Sales People Working. Complete parts a through h below.

Number of ales peaople working	sale (in $1000)
2	10
3	12
5	14
8	15
10	19
10	21
13	21
14	22
17	22
19	27
x(bar)= 10.1	y(bar)=18.3
SD(x)=5.74	SD(y)=5.33

A) find the slope etimate, b₁. B) What does b₁ mean, in thi context? C) Find the intercept, b₀. D) what does b₀ mean in the context Is it meaningful? E) Write the equation that predicts Sales from Number of People Working F) If 18 people were working, what sales do you predict? G) If sales were actually $25,000, what' the value of the residual? H) is this an overestimate or an underestimate?

Answer 1

Cov(x, y) = 29.522, correlation between x and y = r_xy = 0.9636

A) So, from the data, the estimate of slope = b₁ = r_xy * SD(Y) /SD(x) = 0.9636*5.33/5.74 = 0.895

B) This means that if the manager wishes to increase(decrease) his number of sales people by 1 person, then his sales will expected to be increased(decreased) by $0.895*1000 = $895

C) The intercept = b₀ = y(bar) - b₁ * x(bar) = 18.3-0.895*10.1 = 9.26

D) This means if there is no sales people working in the shop, then his estimated sales will be $9.26*1000 = $926

Well, if there is no sales people working, then practically there should not be any sales. So, in that sense, it might not be meaningful.

E) Sales (in $1000) = 9.26 + 0.895 * Number of sales people working

F) As the sales value were reported as multiple of $1000, we should invoke x = 18 in the above equation, then, we get,

Sales (in $1000) = 9.26 + 0.895*18 = 25.37

So, the estimate sales = $25.37*1000 = $25,370

So, the residual = Actual - Estimated = $25,370 - $25,000 = $370

H) As the estimated sale is higher than the actual sale, it is an overestimate.