Question

The data below are the heights of fathers and sons (inches). There are 8 rows in total.

Father Son

44	44
47	47
43	46
41	42
45	47
44	44
44	45
44	45

1. Which statistical test would you use to determine if there is a tendency for tall fathers to have tall sons and short fathers to have short sons? Test for the statistical significance.

2. Compute the regression equation for predicting sons' heights from fathers' heights.

3. Use the equation from #2 to predict the height of a son whose father is 46 inches tall.

4. Should you use the regression equation to predict the hight of a son whose father had a height of 25" when he was the same age as his son?

5. Which statistical test would you use to determine if generations get taller. The question is, are sons taller than their fathers were at the same age?

Answer 1

1. Which statistical test would you use to determine if there is a tendency for tall fathers to have tall sons and short fathers to have short sons? Test for the statistical significance.

Solution:

Here, we have to use t-test for significance of correlation coefficient.

H₀: ρ = 0 versus H_a: ρ ≠ 0

We assume α = 0.05

Test statistic is given as below:

t = r*sqrt(n – 2)/sqrt(1 – r^2)

From given data, we have

r = 0.8, n = 8

t = 0.8*sqrt(8 - 2)/sqrt(1 - 0.8^2)

t = 3.26599

P-value = 0.01712

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that there is a statistically significant relationship exists between the fathers height and sons height.

2. Compute the regression equation for predicting sons' heights from fathers' heights.

The required regression model for the prediction of sons’ heights from fathers’ heights is given as below:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.8
R Square	0.64
Adjusted R Square	0.58
Standard Error	1.095445115
Observations	8
ANOVA
	df	SS	MS	F	Significance F
Regression	1	12.8	12.8	10.66666667	0.01712
Residual	6	7.2	1.2
Total	7	20
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	9.8	10.7847114	0.908693764	0.398540245	-16.58923809	36.18923809
Father	0.8	0.244948974	3.265986324	0.01712	0.200631453	1.399368547

Required regression equation is given as below:

Sons height = 9.8 + 0.8*Fathers height

3. Use the equation from #2 to predict the height of a son whose father is 46 inches tall.

Fathers height = 46

Sons height = 9.8 + 0.8*Fathers height

Sons height = 9.8 + 0.8*46

Sons height = 46.6

4. Should you use the regression equation to predict the hight of a son whose father had a height of 25" when he was the same age as his son?

No, we should not use the regression equation for prediction of the height of a son whose father had a height of 25’’ when he was the same age as his son; because the value 25’’ is not lies between the range of fathers heights.

5. Which statistical test would you use to determine if generations get taller. The question is, are sons taller than their fathers were at the same age?

WE will use the one tailed dependent samples t test or paired t test for checking whether the average difference between the sons height and fathers height is positive or not.