Question

a.	In the region to the right, produce a scatterplot of the height versus footlength data (remember this means footlength runs along the horizontal axis as the independent variable and height along the vertical axis as the dependent variable). Based upon your scatterplot, briefly discuss below your thoughts on whether the “visual” trend between the individuals’ footlength and height appears linear, curvilinear, or has no general trend at all.
b.	Complete the following:
	i.	Include the trend line's graph and equation (least squares line) on the scatterplot created in part a. Give the line's equation below and explain within this context what the "x" and "y" variables represent in the equation.
	ii.	Below, explicitly state the slope of your trend line and discuss what the value of the slope signifies in terms of this context.
c.	Determine the value of the correlation coefficient (r) for this paired data. Explain what this value tells you regarding these two variables. Determine the value of the coefficient of determination (r^2) for this paired data. Explain what this value tells you regarding these two variables.

Male	23.0	162.5
Female	21.5	167.5
Male	33.0	193.0
Female	24.5	165.0
Female	24.0	160.0
Female	23.5	162.5
Male	28.0	189.0
Female	24.5	165.0
Female	24.5	164.0
Female	27.0	175.0
Male	25.0	177.0
Male	28.0	188.0
Female	20.0	167.5
Female	25.0	163.0
Male	26.5	181.0
Female	27.0	183.0
Female	23.5	169.0
Female	25.5	161.5
Male	26.0	184.0
Female	23.5	167.5
Female	28.0	167.5
Male	26.5	171.5
Female	23.5	170.0
Female	25.0	175.0
Male	25.5	173.0
Male	30.5	180.5
Female	23.0	160.0
Female	22.0	172.5
Male	29.0	180.0
Female	21.5	173.0
Male	26.5	176.5
Male	30.5	184.0
Female	22.5	154.5
Female	25.0	162.5
Female	25.0	162.5
Female	24.0	170.0
Female	23.0	167.5
Male	28.0	179.0
Female	18.0	166.0
Female	23.0	160.0
Female	24.0	170.5
Female	27.0	177.0
Female	23.0	167.0
Male	27.5	190.5
Male	25.5	167.0
Female	16.5	152.0
Female	24.0	152.5
Female	24.0	163.0
Female	24.0	157.0
Female	25.5	167.0
Female	23.5	161.0
Female	25.5	161.0
Female	26.0	158.0
Female	23.5	152.5
Female	25.0	169.0
Female	25.5	165.0
Male	25.0	172.5
Female	25.5	170.5
Male	29.0	186.0
Female	23.0	157.5
Male	28.5	180.5
Male	28.0	177.5
Male	30.0	185.5
Female	24.0	155.0
Male	29.5	185.0
Male	28.0	181.0
Female	22.0	154.0
Female	20.0	170.0
Female	23.0	159.0
Female	31.0	165.0
Male	25.5	186.0
Male	29.0	180.0
Female	23.0	148.0

Answer 1

a) ### By using R command:

> footlength=c(23,21.5,33,24.5,24,23.5,28,24.5,24.5,27,25,28,20,25,26.5,27,23.5,25.5,26,23.5,28,26.5,23.5,25,25.5,30.5,23,22,29,21.5,26.5,30.5,22.5,25,25,24,23,28,18,23,24,27,23,27.5,25.5,16.5,24,24,24,25.5,23.5,25.5,26,23.5,25,25.5,25,25.5,29,23,28.5,28,30,24,29.5,28,22,20,23,31,25.5,29,23)
> footlength
[1] 23.0 21.5 33.0 24.5 24.0 23.5 28.0 24.5 24.5 27.0 25.0 28.0 20.0 25.0 26.5
[16] 27.0 23.5 25.5 26.0 23.5 28.0 26.5 23.5 25.0 25.5 30.5 23.0 22.0 29.0 21.5
[31] 26.5 30.5 22.5 25.0 25.0 24.0 23.0 28.0 18.0 23.0 24.0 27.0 23.0 27.5 25.5
[46] 16.5 24.0 24.0 24.0 25.5 23.5 25.5 26.0 23.5 25.0 25.5 25.0 25.5 29.0 23.0
[61] 28.5 28.0 30.0 24.0 29.5 28.0 22.0 20.0 23.0 31.0 25.5 29.0 23.0
> height=c(162.5,167.5,193,165,160,162.5,189,165,164,175,177,188,167.5,163,181,183,169,161.5,184,167.5,167.5,171.5,170,175,173,180.5,160,172.5,180,173,176.5,184,154.5,162.5,162.5,170,167.5,179,166,160,170.5,177,167,190.5,167,152,152.5,163,157,167,161,161,158,152.5,169,165,172.5,170.5,186,157.5,180.5,177.5,185.5,155,185,181,154,170,159,165,186,180,148)
> height
[1] 162.5 167.5 193.0 165.0 160.0 162.5 189.0 165.0 164.0 175.0 177.0 188.0
[13] 167.5 163.0 181.0 183.0 169.0 161.5 184.0 167.5 167.5 171.5 170.0 175.0
[25] 173.0 180.5 160.0 172.5 180.0 173.0 176.5 184.0 154.5 162.5 162.5 170.0
[37] 167.5 179.0 166.0 160.0 170.5 177.0 167.0 190.5 167.0 152.0 152.5 163.0
[49] 157.0 167.0 161.0 161.0 158.0 152.5 169.0 165.0 172.5 170.5 186.0 157.5
[61] 180.5 177.5 185.5 155.0 185.0 181.0 154.0 170.0 159.0 165.0 186.0 180.0
[73] 148.0
> plot(footlength,height,xlab="footlength",ylab="height",main="Scatterplot")
>

From the above scatter plot it is clear that there is a strong positive linear relationship between foot length and height.

b) ### Using R command:

> fit=lm(height~footlength)
> fit

Call:
lm(formula = height ~ footlength)

Coefficients:
(Intercept) footlength  
109.503 2.394  

The equation of line of best fit is:

> abline(fit, col="red") ### regression line

Here the slope of the regression line is =2.394 which indicates that a unit increase in the foot length results in the increase in the height of a person by 2.394 unit.

c) The correlation coefficient is r=0.67

## By using command:

> cor(footlength,height)
[1] 0.6669175

The correlation coefficient tells us that there is a strong positive correlation between foot length and height.

The value of coefficient of determination is

Which tells us that the about 45% variability in the height is explained by the foot length.

Overall the model well fits the data.