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

a-1. Calculate the sample correlation coefficient between age and happiness. (Round your answer to 2 decimal places.)

a-2. Interpret the sample correlation coefficient between age and happiness.

• The correlation between age and happiness is positive and moderate.

• The correlation between age and happiness is negative and moderate.

• There is no correlation between age and happiness.

b. Draw a scatterplot to evaluate your interpretation of the calculated sample correlation value.

• The interpretation is reliable given that for the majority of observations, a direct relationship exists.

• The interpretation is reliable given that for the majority of observations, an inverse relationship exists.

• The interpretation is flawed given the linear relationship that seems to exist between these two variables.

• The interpretation is flawed given the non-linear relationship that seems to exist between these two variables.

Answer 1

Sol:

a-1)

(х-х) (у-у) 62.000 49.00 43.8906 100.0000 66.2500 66.000 2 21.3906 36.0000 27.7500 67.000 213.8906 25.0000 73.1250 71.000 87.000 4 87.8906 9.3750 84.00 805.1406 225.0000 425.6250 60.000 213.8906 144.0000 175.5000 86.000 83.00 749.3906 196.0000 383.2500 18.00 78.000 1415.6406 36.0000 225.7500 59.000 385.1406 169.0000 255.1250 61.00 63.000 28.8906 -48.3750 77.000 15.00 1650.3906 25.0000 90.000 86.00 324.0000 12 922.6406 546.7500 73.00 70.000 301.8906 34.7500 62.000 558.1406 100.0000 236.2500 93.000 15 805.1406 441.0000 595.8750 72.000 1064.3906 0.0000 52.00 58.000 13.1406 196.0000 50.7500 73.000 72.00 18 268.1406 16.3750 63.000 81.0000 19 54.3906 66.3750 30.00 66.000 20 656. 6406 36.0000 153.7500 72.00 78.000 268.1406 36.0000 98.2500 47.00 60.000 74.3906 144.0000 103.5000 95.000 88.00 529.0000 1048.1406 744.6250 69.00 72.000 178.8906 0.0000 1335 1728 11829.6250 2930.0000 3365.0000 Mean 55.6250 72.0000 correlation coefficientr Sxy/(SSx*Ssy)A1/2 0.5716 3.2671 test statistic: P value 0.0035 у 0.2845x 56.177 R2 0.3267 100.000 90.000 80.000 70.000 60.000 50.000 40.00 60.00 80.00 100.00

Sample correlation coefficient =0.57

a-2)

Correlation coefficient is

VR2-V0.3267= 0.5716 r

Therefore, the correlation between age and happiness is positive and moderate.

b)

   The scatterplot is:

100 90 y 0.2845x + 56.177 R2 0.3267 80 70 60 a 50 E 40 30 20 10 20 40 60 80 100 Age

The correct option is:

The interpretation is flawed given the non-linear relationship that seems to exist between these two variables.