Problem 3 - Consumer behaviour theory(16 marks)

"Consumers are likely to purchase more of a good when the price of the good decreases, and vice versa."

As stated above, a marketing theory in the field of consumer behaviour believes that an indirect relation is likely to exist between the price and the number of units purchased for most of the goods and services.In examining this, you were commissioned as a student in the Business Data Analysis course to conduct a mock market research surveying a sample of potential customers in purchasing a hypothetical new product.You randomly selected 16 of your fellow students as the potential customers for the product.You questioned them on the number of units they were likely to purchase for a given price that you set.The data you collected from the mock survey are displayed below.You are tasked in performing the relevant inferential analyses on the relation between the two concerned variables by addressing the below set of questions.

Set Price (x) & Number of units purchased in $ (y)

X = 13 15 13.5 15 16.5 13.5 14 14.2 16.5 17 17.2 18 20 21 19 19.5

y = 21 15 21 13 14 22 19 17 13 13 14 15 17 14 17 16

a)Which of the two concerned variables is the dependent variable?Which one is the independent variable?Give an explanation of the rationale of the selection of the dependent and independent variables.(1 mark)

b)The following sums have been computed:

summation xi = 262.8 & summation yi = 261 & summation xiyi = 4227.2 & summation xi^2 = 4416.73 & summation yi^2 = 4395

Use this info to calculate S_x², S_y² and S_xy.Display working.

a)Calculate the sample correlation coefficient.Display working.Then, provide an interpretation of the calculated correlation in terms of the relation between price and number of units purchased.(2 marks)

b)Calculate the slope coefficient of the sample linear regression.Then, calculate the intercept coefficient of the sample linear regression. Display working.(1 mark)

c)Write the estimated sample linear regression equation. (1 mark)

d)Provide interpretations of the slope coefficient and the intercept coefficient of the sample linear regression in terms of the relation between price and number of units purchased.

(2.5 marks)

e)Predict the number of units purchased if price is $25.Display working.Comment on the validity of this prediction.(2 marks)

f)Calculate the coefficient of determination for the regression line.Provide an interpretation of the calculated coefficient of determination in terms of the relation between price and number of units purchased.(2 marks)

g)Perform the relevant hypothesis test at the 5% level of significance to see if a negative relation exists between price and number of units purchased.Display working of the six-steps of the hypothesis test.The t test-statistic has been calculated.t calculated = -2.35.

(3 marks)