Question

Microeconomic theory states that an inverse relation is likely to exist between price and quantity demanded for normal goods. To examine the validity of this theory in the market of apples, a market researcher observed the daily market price of a box of apples and the quantity of boxes of apples demanded in a local wholesaler fruit market over a period of nine days. By referring to data and additional information displayed below, assist the researcher in performing correlation and simple linear regression analyses on the two variables.  

day	price in $	quantitydemanded in thousands of boxes
1	13	33.5
2	11.5	34.5
3	9.5	39.2
4	13.5	31.3
5	14.3	32.25
6	15	34.3
7	16	32.35
8	17.5	31.25
9	18.2	27.5

∑?? = 128.5     ∑? ? = 295.9   ∑??2 = 1896.73   ∑? ? 2 = 9808.03    ∑?? ? ? = 4164.6

a) State the dependent and independent variables. Briefly explain your selection of the dependent and independent variables.         

b) Calculate Sx2, Sy2 and Sxy using the information given above. Display working.

   c) Calculate the sample correlation coefficient. Display working.               

d) Provide an interpretation of the calculated value of the sample correlation coefficient in terms of the relation between price and quantity demanded.                                       

e) Calculate the slope coefficient of the sample linear regression equation. Display working.

f) Provide an interpretation of the slope coefficient you calculated in terms of the relation between price and quantity demanded.                                          

g) Calculate the intercept coefficient of the sample linear regression equation. Display working.               

h) Provide an interpretation of the intercept coefficient you calculated in terms of the relation between price and quantity demanded.                                                                      

i) State the estimated sample linear regression equation.            

j) Predict the quantity demanded if price is $19. Display working. Comment on the validity of this prediction.                      

k) Conduct a test on the slope coefficient to see if a negative relation exists between the two variables. Use a 5% level of significance. Display working of the six steps hypothesis test.   The t test-statistic has been calculated. It equals -4.40.                             

l) Calculate the coefficient of determination for the regression line.                   

m) Provide an interpretation of the calculated coefficient of determination in terms of the relation between price and quantity demanded.                   

Answer 1

a) State the dependent and independent variables. Briefly explain your selection of the dependent and independent variables.     

The dependent variable is the quantity demanded and the independent variable is the price. Since the quantity demanded in general depends on the price, I had Chosen the independent and dependent variables.

∑?? = 128.5     ∑? ? = 295.9   ∑??2 = 1896.73   ∑? ? 2 = 9808.03    ∑?? ? ? = 4164.6

b). b) Calculate Sx2, Sy2 and Sxy using the information given above. Here, n=9.

  

b)

c) Calculate the sample correlation coefficient.

Correlation coefficient is given by:

  

The correlation coefficient r=-0.8572

  d) Provide an interpretation of the calculated value of the sample correlation coefficient in terms of the relation between price and quantity demanded.      

The correlation between the quantity demanded and price is -0.8572 which indicates that when there is an increase in price, the demand decreases and vice versa. The correlation is quite close to -1 which shows a linear model is a good predictor for price and demand in this case.

e) Calculate the slope coefficient of the sample linear regression equation.

The slope is given by

  

f) Provide an interpretation of the slope coefficient you calculated in terms of the relation between price and quantity demanded.   

Teh slope here is -0.9703, which implies that whenever, there is an increase(decrease) of 1$ in price, the quantity demanded will be reduced by 0.9703 units or ~970.3 boxes.

f) Provide an interpretation of the slope coefficient you calculated in terms of the relation between price and quantity demanded.   

The intercept is given by

  

h) Provide an interpretation of the intercept coefficient you calculated in terms of the relation between price and quantity demanded.

The intercept is 42.7315 which is the quantity demanded when the price is 0$. That is when the price is 0$, the quantity demanded is 42732 boxes.

i) State the estimated sample linear regression equation.   

The estimated regression equation is

j) Predict the quantity demanded if price is $19. Display working. Comment on the validity of this prediction.

The estimated quantity is 24.2958. It is to be noted that this estimated quantity is quite as expected if we extrapolate it further. At times, this has to be interpreted with caution since the price is out of the range.

k) Conduct a test on the slope coefficient to see if a negative relation exists between the two variables. Use a 5% level of significance. Display working of the six steps hypothesis test.   The t test-statistic has been calculated. It equals -4.40.       

1) Null hypothesis:

2) Alternate Hypothesis:

3) Level of significance:

4). We need to have the estimate of and its standard error.

5). Test statistic :

6). The rejection region is given by

7). Since the t -test statistic is -4.40 and which falls in the rejection region, we reject the null hypothesis and conclude that the regression coefficient cannot be assumed to be 0.

l) Calculate the coefficient of determination for the regression line.   

The coefficient of determination

  

m) Provide an interpretation of the calculated coefficient of determination in terms of the relation between price and quantity demanded.        

Since , which indicates that the linear regression model could explain 73.48% of teh total variation in the data.