Question

A study was conducted to analyze whether there is a linear relation between the number of customers who visited a store and the total sales at that store. 

The following data was collected over a period of 14 days

(a) What is the dependent and independent variable? 

(b) Prepare a scatter plot. What is the direction of correlation?

(c) What is the equation of the regression line? 

d) What is the coefficient of determination (r) and interpret the meaning of it. 

(e) Calculate the correlation coefficient (r) and interpret the meaning of it. 

(f) Predict total sales if 120 customers visited the store. 

(g) Would it be possible to use this regression model to predict the Total Sale for 500 customers? Why? 

Answer 1

#simple linear regression

x-number of customers

y-total sales

a)

x-independent variable

y- dependent variable

> x=scan("clipboard");x

Read 14 items

[1] 125 88 111 121 102 98 106 133 128 105 118 127 144 136

> y=scan("clipboard");y

Read 14 items

[1] 35.1 28.5 30.3 36.2 33.4 30.2 31.2 44.5 39.4 32.5 33.2 40.5 58.2 41.3

> reg=lm(y~x);reg

Call:

lm(formula = y ~ x)

Coefficients:

(Intercept)            x

   -11.9734       0.4154

> summary(reg)

Call:

lm(formula = y ~ x)

Residuals:

    Min     1Q Median      3Q     Max

-4.8547 -2.9415 -0.5736 1.4018 10.3522

Coefficients:

             Estimate Std. Error t value Pr(>|t|)   

(Intercept) -11.97342    8.47484 -1.413    0.183   

x             0.41542    0.07163   5.799 8.48e-05 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.161 on 12 degrees of freedom

Multiple R-squared: 0.737,     Adjusted R-squared: 0.7151

F-statistic: 33.63 on 1 and 12 DF, p-value: 8.482e-05

c) hence regression of line is

y=   -11.9734 + 0.4154*x

d) coefficient of determination (r^2) is

r^2=0.737

interpitation   73.7 % of variation in y is explain by x

                      and 26.3 % is un explain

d) correlation coefficient r

>r= cor(x,y);r

[1] 0.8585037

r>0 implies that the relation between x&y is positive correlation