Question

Many attempts have been made to relate happiness with various factors. One such study relates happiness with age and finds that holding everything else constant, people are least happy when they are in their mid 40's ( The Economists December 16, 2010). The accompanying data shows a respondent's age and his or her perception of well being on a scale from 0 to 100.

pt1

1) Calculate and interpret covariance using Excel (10 points)

2)Use Excel to calculate and interpret the correlation coefficient between age and happiness. (10 points)

pt2

3)Use Excel to estimate a simple regression model with happiness as the response variable and age as the explanatory variable. (Excel product should look similar to p519) (10 points)

4) Construct the simple regression model (the equation) and round to two digits after the decimal for the y-intercept and slope.a) Interpret the slope coefficient. b) Use the estimate to predict happiness when age equals 25,45,60. (Show work typed)(10 points)

5)Find the coefficient of determination (R²) on the simple regression model and interpret the output about the data happiness and age? (p530-531) (5 points)

Age	Happiness
51	62
53	66
43	67
67	71
86	87
43	60
85	86
18	78
37	59
60	63
17	77
86	90
75	70
32	62
84	93
23	72
51	58
72	73
65	63
30	66
73	78
46	60
89	95
70	72

Answer 1

a)

The covariance comes out to be 147.6. It signifies the relationship between the movement of two variables, which are age and happiness here.

b)

The correlation comes out to be 0.6, which shows a medium positive correlation between the two variables.

c)

We will be applying the Linear regression model here, it can be done by using the function =LINEST(y_value, x_value, TRUE, TRUE) where y_values contain values of Happiness here and x_values have Age values.

Select 5 rows and 2 columns and then write the formula in the first cell and after that, press Shift + Ctrl + Enter.    

The equation comes out to be -
Happiness = 56 + 0.28*Age  

d)

For every unit increase in the age, which is for every year increase in age, happiness increases by 0.28 units.

When Age = 25, Happiness = 56 + 0.28*25 = 63.08

When Age = 45, Happiness = 56 + 0.28*45 = 68.75

When Age = 60, Happiness = 56 + 0.28*60 = 72.99

e)

The coefficient of determination comes out to be 0.33, which means that the change in happiness can be 33 percent explained by the age.

A C D E F H 1 K L M N o P covariance correlation 147.6 COVARIANCE.S(A2:A25,B2:B25) 0.6 CORREL(A2:A25,B2:B25) Age 51 53 43 67 86 43 85 18 37 60 17 86 75 32 84 23 51 72 65 30 73 B Happiness 62 66 67 71 87 60 86 78 59 63 77 90 70 62 93 72 58 73 63 66 78 slope intercept 0.283022 56.00926 0.0864 5.249746 0.327842 9.461461 10.73039 22 960.5767 1969.423 2 r**2 3 4 UI US 7 3 Happiness = 56 +0.28*Age 1