Question

You want to determine if there is a linear relationship between mental health and physical health. You have several people assessed by a psychologist and a medical doctor. The psychologist assess mental health (ment) on a scale of 1-30 (higher numbers mean better mental health) and the medical doctors rated their overall physical health (phys) from 1-30 (higher numbers mean better health). You obtain the data below. (note: there is more information than you need)

Person	MENT	PHYS	+ -	Rank   MENT	Rank   PHYS	MENT*PHYS	MENT2	PHYS2
1	16	17	+	6	7	272	256	289
2	17	16	-	5	8	272	289	256
3	12	20	+	9	5	240	144	400
4	18	23	+	4	3	414	324	529
5	19	25	+	3	2	475	361	625
6	20	21	+	2	4	420	400	441
7	15	19	+	7	6	285	225	361
8	14	15	+	8	9	210	196	225
9	23	28	+	1	1	644	529	784
10	11	14	+	10	10	154	121	196
MEAN	16.5	19.8		5.5	5.5	338.6	284.5	410.6
Variance	13.61111	20.62222		231	231	863867.26	609935.62	1273662.67

Determine the equation for the least squares regression line predicting mental health from physical health.

ΣY =

ΣX =

ΣY² =

ΣX² =

ΣXY =

SS_XY =

SS_X =

SS_Y =

N =

b =

mean of X =

mean of Y =

a =

type your final regression line in this box

What distance above and below the regression line would capture approximately 96% of the data points

If someone's physical health score was 15 what would you predict for their mental health score?   

What is the correlation between mental health ad physical health   

How much variance in mental health can be explained by physical health?   

Explain this correlation in plain terms that someone who knew nothing about statistics would understand   

Answer 1

Person	PHYS (X)	MENT (Y)	X^2	XY	Y^2
1	17	16	289	272	256
2	16	17	256	272	289
3	20	12	400	240	144
4	23	18	529	414	324
5	25	19	625	475	361
6	21	20	441	420	400
7	19	15	361	285	225
8	15	14	225	210	196
9	28	23	784	644	529
10	14	11	196	154	121
SUM	198	165	4106	3386	2845

Observation	Predicted Y	Residuals	Resdual ^2
1	14.70474138	1.295258621	1.677694894
2	14.06357759	2.936422414	8.622576592
3	16.62823276	-4.628232759	21.42053847
4	18.55172414	-0.551724138	0.304399524
5	19.83405172	-0.834051724	0.695642279
6	17.26939655	2.730603448	7.456195192
7	15.98706897	-0.987068966	0.974305143
8	13.42241379	0.577586207	0.333605826
9	21.7575431	1.242456897	1.54369914
10	12.78125	-1.78125	3.172851563
SUM			46.20150862

Interpretation of correlation: The correlation coefficient tells the strength of association between two values that vary with each other. In this case, the physical and mental health ratings by the doctors are assessed independently, and then we try to ascertain if one can be predicted from the other. Correlation is measured from -1 to +1, where -1 would indicate a strong negative linear relation i.e. one decreases as the other increases, 0 would mean no correlation and +1 would mean a strong (perfect) positive linear relationship.

Link to excel on which values were obtained: https://drive.google.com/file/d/1iXtbYBXIUeS5BKz1zo6jl3j_zYD0x8fA/view?usp=sharing