Question

The following question examines the prices of televisions. In this question, we’re going to consider regressions of the form:

pricei = β0 + β1sizei + β2size2i + β3UHDi + εi

Where pricei is the retail price of the television in dollars, sizei is the size of the TV screen in inches, and UHDi is a binary {0,1} indictor for whether or not the television supports ultra-high definition (versus conventional high definition). We run the regression two different ways. The coefficients are printed in the table below, with standard errors in parentheses.

	Model 1	Model 2
size	75.84 (18.92)	-282.6 (51.86)
size2		4.367 (0.769)
UHD	-50.41 (140.4)	164.2 (60.68)
Cons.	-1902 (614.8)	5001 (858.3)
N	98	98
R2	.510	.708

1. a) In dollars, how much more expensive is a 42-inch TV than a 32-inch TV according to Model 1?

2. b) In dollars, how much more expensive is a 42-inch TV than a 32-inch TV according to Model 2?

3. c) Using the results from Model 2, construct a 95% confidence interval around the effect of a TV supporting ultra-high definition on its price. [3 points]

4. d) Evaluate the null hypothesis that supporting ultra-high definition does not affect a television’s price according to both Model 1 and Model 2. Use a 95% significant level. Does your conclusion change based on which model you examine?

5. e) Based on the table above, do you think the relationship between a TV’s screen size and price is linear? Why or why not? [3 points]

Answer 1

(a) For Model 1

The price of a 42 inch TV is estimated using the regression equation given in the question.

Here size =42 ,then the average price of the TV is

p1=75.84*42-11902=1283.28 dollar.

Also price of a 32 inch TV is estimated using the regression equation given in the question.

Here size =32 ,then the average price of the TV is

p2=75.84*32-11902=524.88 dollar.

So p1-p2=758.4 dollar

Hence the 42 inch TV is 758.4 dollar more expensive than that of 32 inch TV according to model 1.

Here we estimated the price of TV not supporting ultra-high definition but the result is same for the television supporting ultra-high definition.

(b) For Model 2

The price of a 42 inch TV is estimated using the regression equation given in the question.

Here size =42 ,then the average price of the TV is

p1= -282.6*42+4.367*(42*42)+5001=835.188 dollar.

Also price of a 32 inch TV is estimated using the regression equation given in the question.

Here size =32 ,then the average price of the TV is

p2=-282.6*32+4.367*(32*32)+5001=429.608 dollar.

So p1-p2=405.58 dollar

Hence the 42 inch TV is 405.58 dollar more expensive than that of 32 inch TV according to model 2.

Here we estimated the price of TV not supporting ultra-high definition but the result is same for the television supporting ultra-high definition.

(c) From the results of model 2, we construct the 95 % confidence interval for ; which is

Here is least square estimate of and S.E.() is standard error of given in the table and (=1.96) is 100(1-alpha)% critical point of standard normal distribution.

So  the 95 % confidence interval for is .

(d) The hypothesis is

H0: against the alternative H1: .

Since the sample size ,N =98, is quit large(>30) so for testing the null hypothesis we use Z-test

The test statistic is given by

; which follows standard normal distribution.

Here is least square estimate of and S.E.() is standard error of given in the table.

The critical value of Z = 1.96 at 5% level of significance.

For model 1, the calculated value of Z is

For model 2, the calculated value of Z is

Since for model 1, the absolute value of Z is less than the tabulated value of Z, hence we cannot reject the null hypothesis at 5% level of significance.

Also for model 2, the absolute value of Z is greater than the tabulated value of Z, hence we reject the null hypothesis at 5% level of significance.

Thus according to the test result based on model 1, we conclude that TV, either supporting or not supporting ultra-high definition ,does not have any significant effect on its price. But if we go for model 2, then the test result, based on it ,shows that TV, either supporting or not supporting ultra-high definition ,does have any significant effect on its price.

(e) To asses whether there is linear relationship between size and price of TV or not, we use F test.

The test statistic is

where N=98 and k=4 and R2=0.51 & 0.708 for model 1 and 2 respectively.

So for model 1, F=49.4388 and  for model 2, F=75.9726.

Here for both the models, value of F statistic is highly significant, so we conclude that the relationship between size and price of TV is linear.