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-40s (The Economist, December 16, 2010). Data are collected on a respondent’s age and his/her perception of well-being on a scale from 0 to 100; the data is presented below.

Age Happiness
49 62
51 66
41 67
65 71
84 87
41 60
83 86
18 78
36 59
61 63
15 77
86 90
73 70
32 62
84 93
23 72
52 58
72 73
63 63
30 66
72 78
47 60
88 95
69 72

Required:
a. Calculate the sample correlation coefficient between age and happiness.
b. Interpret the sample correlation coefficient between age and happiness.
c. Draw a scatterplot to evaluate your interpretation of the calculated sample correlation value.

Answer 1

a) 0.572

b) There is moderate positive relationship between age and happiness.

c)

Scatter plot is attached

Explanation:

a)

`Sample correlation coefficient=r=(nsumxy-(sumx)(sumy))/([nsumx^(2)-(sumx)^2][ nsumy^(2)-(sumy)^2] )`

Age(x) Happiness(y) x² y² xy

49 62 2401 3844 3038

51 66 2601 4356 3366

41 67 1681 4489 2747

65 71 4225 5041 4615

84 87 7056 7569 7308

41 60 1681 3600 2460

83 86 6889 7396 7138

18 78 324 6084 1404

36 59 1296 3481 2124

61 63 3721 3969 3843

15 77 225 5929 1155

86 90 7396 8100 7740

73 70 5329 4900 5110

32 62 1024 3844 1984

84 93 7056 8649 7812

23 72 529 5184 1656

52 58 2704 3364 3016

72 73 5184 5329 5256

63 63 3969 3969 3969

30 66 900 4356 1980

72 78 5184 6084 5616

47 60 2209 3600 2820

88 95 7744 9025 8360

69 72 4761 5184 4968

sumx=1335

sumy=1728

sumx²=86089

sumy²=127346

sumxy=99485

n=24

`Sample correlation coefficient=r=(24(99485)-(1335)(178))/([24(86089)-(1335)^2][ 24(127346)-(1728)^2] )`

Sample correlation coefficient=0.572

b)

Sample correlation coefficient=0.572 shows the moderate positive correlation between age and happiness. As the age increases happiness also moderately increases.

c)

The attached scatter plot shows that the as age increases happiness also moderately increases.