Question

1. A business maintains a centralized warehouse for the distribution of products ordered via the internet.  Management is examining the relation between distribution costs and the number of orders processed in a month.  The goal is to estimate monthly cost from the number of orders.  Here is the data for the past twelve months, with Cost given in thousands of dollars and Orders given in the thousands.  Orders are the explanatory variable and Cost is the dependent variable.

Cost	52.9	71.7	85.6	63.7	72.8	68.4	52.5	70.8	82.0	74.4	70.8	54.1
Orders	4.02	3.81	5.31	4.26	4.30	4.10	3.21	4.81	5.24	4.73	4.41	2.92

1. Find the equation of the least square regression line.
2. Predict the monthly cost for 5000 orders.
3. Draw a scatter plot of the data andplot the regression line on the scatter plot.

   d.  Find r and describe the relationship.

   e. Find r²and summarize what is means.

f.  Test at the 5% level of significance, the null hypothesis ₁=0 against a suitable hypothesis about linear correlation.   State the hypothesis, the p-value of the test and state your conclusion.

g.  Find the 95% confidence interval for the predicted value in part b.

Answer 1

a.

Sum of X = 51.12
Sum of Y = 819.7
Mean X = 4.26
Mean Y = 68.3083
Sum of squares (SS_X) = 5.7942
Sum of products (SP) = 74.156

Regression Equation = ŷ = bX + a

b = SP/SS_X = 74.16/5.79 = 12.798

a = M_Y - bM_X = 68.31 - (12.8*4.26) = 13.788

ŷ = 12.798X + 13.788

b. For x=50, ŷ = (12.798*50) + 13.788=653.688

c.

d.

X Values
∑ = 51.12
Mean = 4.26
∑(X - M_x)² = SS_x = 5.794

Y Values
∑ = 819.7
Mean = 68.308
∑(Y - M_y)² = SS_y = 1278.109

X and Y Combined
N = 12
∑(X - M_x)(Y - M_y) = 74.156

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 74.156 / √((5.794)(1278.109)) = 0.8617

As r is near to 1 and it is positive so there is strong positive correlation

e. So r^2=0.8617^2=0.7425

So 74.25% of variation in cost is explained by orders.