Question

A multiple regression model is to be constructed to predict the final exam score of a university student doing a particular course based upon their mid-term exam score, the average number of hours spent studying per week and the average number of hours spent watching television per week.

Data has been collected on 30 randomly selected individuals: hide data

Download the data

Final score	Mid-term Score	Hours studying per week	Hours watching TV per week
76	85	19	34
60	85	3	11
42	63	10	33
32	40	6	12
46	65	6	16
48	72	13	30
30	37	14	33
47	47	9	26
33	26	19	7
60	65	18	8
59	79	13	24
28	29	10	22
24	33	7	9
59	77	5	6
66	93	17	7
51	48	18	29
74	98	4	7
29	23	8	15
31	31	6	9
69	79	18	15
60	73	3	6
62	89	11	31
49	52	19	6
37	44	14	9
63	94	10	26
62	89	7	27
30	31	18	32
42	60	14	17
54	70	4	22
73	97	19	28

a)Find the multiple regression equation using all three explanatory variables. Assume that X₁ is mid-term score, X₂ is hours studying per week and X₃ is hours watching television per week. Give your answers to 3 decimal places.

y^ =  + mid-term score + hours studying + hours watching television

b)At a level of significance of 0.05, the result of the F test for this model is that the null hypothesis isis not rejected.

For parts c) and d), using the data, separately calculate the correlations between the response variable and each of the three explanatory variables.

c)The explanatory variable that is most correlated with final score is:

mid-term score
hours studying per week
hours watching television per week

d)The explanatory variable that is least correlated with final score is:

mid-term score
hours studying per week
hours watching television per week

e)The value of R² for this model, to 2 decimal places, is equal to

f)The value of s_e for this model, to 3 decimal places, is equal to

g)Construct a new multiple regression model by removing the variable average hours spent watching television per week. Give your answers to 3 decimal places.

The new regression model equation is:

y^ =  + mid-term score + hours studying

h)In the new model compared to the previous one, the value of R² (to 2 decimal places) is:

increased
decreased
unchanged

i)In the new model compared to the previous one, the value of s_e (to 3 decimal places) is:

increased
decreased
unchanged

Answer 1

a. The regression line is

y= 8.205 + 0.617 X1 + 0.501 X2 - 0.141 X3

b. At a level of significance of 0.05, the result of the F test for this model is that the null hypothesis is rejected.

c. Since the correlation between Final scores and Mid-term score is highest(i,e 0.931), Thus The explanatory variable that is most correlated with the final score is mid-term scores.

d. Since the correlation between Final scores and the Hours watching TV per week is lowest(i,e 0.060), Thus The explanatory variable that is least correlated with the final score is Hours watching Tv per week

e. The value of R² for this model, to 2 decimal places, is 0.90

f. The value of s_e for this model, to 3 decimal places, is 5.226

g. The required regression line is

y^ = 6.812+ 0.610 mid-term score + 0.435 hours studying

h. In the new model compared to the previous one, the value of R² = 0.89 and  is decreased

i) In the new model compared to the previous one, the value of s_e = 5.320 and is increased

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