Question

(We are using the same data set we used in Question 2)

You are the manager of the supermarket on the ground floor of Holmes Building. You are wondering if there is a relation between the number of students attending class at Holmes Institute each day, and the amount of chocolate bars sold. That is, do you sell more chocolate bars when there are a lot of Holmes students around, and less when Holmes is quiet? If there is a relationship, you might want to keep less chocolate bars in stock when Holmes is closed over the upcoming holiday. With the help of the campus manager, you have compiled the following list covering 7 weeks:

Page 4 of 9

Weekly attendance 472

413

503

612

399

538

455

Number of chocolate bars sold 6916

5884

7223

8158

6014

7209

6214

Tasks:

a. Calculate AND interpret the Regression Equation. You are welcome to use Excel to check your

calculations, but you must first do them by hand. Show your workings.

(Hint 1 - As manager, which variable do you think is the one that affects the other variable? In other words, which one is independent, and which variable’s value is dependent on the other variable? The independent variable is always x.

Hint 2 – When you interpret the equation, give specific examples. What happens when Holmes are closed? What happens when 10 extra students show up?)

b. Calculate AND interpret the Coefficient of Determination.

Answer 1

I can't do it by hand, but I can do it in excel and show you the output.

a. Calculate AND interpret the Regression Equation. You are welcome to use Excel to check your calculations, but you must first do them by hand. Show your workings.

As manager, which variable do you think is the one that affects the other variable? In other words, which one is independent, and which variable’s value is dependent on the other variable? The independent variable is always x.

Attendance is the variable that impacts the chocolates sold. Hence, attendance is the independent variable and chocolates sold is the dependent variable. Hence, X is Attendance and Y is Number of chocolate bars sold.

The standard equation of regression will be:

Hence, Y = aX + b

What happens when Holmes are closed?

When Homes are closed, X i.e. attendance will be zero. Hence number of chocolate bars sold = Y = b. Hence "b" represents the number of chocolates that will be sold even if Holmes is closed.

What happens when 10 extra students show up?

The coefficient of X that is "a" shows the sensitivity of Y with respect to X. That is, if X changes be 1 unit, Y will change by a units. Thus, if 10 extra students turn up, X will increase by 10 and hence Y = number of chocolate bars sold will increase by 10a.

b. Calculate AND interpret the Coefficient of Determination.

Summary of regression output as produced by excel is shown here:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.967992639
R Square	0.93700975
Adjusted R Square	0.9244117
Standard Error	224.5951736
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	1	3751816.754	3751816.754	74.37736352	0.000346012
Residual	5	252214.9601	50442.99201
Total	6	4004031.714
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept (b)	1628.688985	605.9000187	2.688049074	0.043399872	71.17340223	3186.20457	71.17340223	3186.20457
X Variable 1 (a)	10.67723382	1.23805051	8.624231184	0.000346012	7.494723664	13.859744	7.494723664	13.859744

Coefficient of determination is given by R square. Please see the table above.

In the output summary, R square = 0.93700975

Interpretation: Coefficient of determination or R square is a measure of (square of) correlation between X and Y variables. A zero value means X and Y are not correlated at all and hence Y can't be predicted based on X. A high value implies high correlation and hence data shows a good fit between X and Y and Y can be predicted based on X.

In this case R square value is very good. A value of 0.93 means a good fit between the data and we can therefore safely say that:

• Attendance is indeed a driver of chocolates sold
• The correlation is positive. That is increase in attendance increases the sale while reduction attendance decreases the sale.
• And future sales can be predicted based on attendance