1. Head width (cm) Biting force (grams/centimer 2 ) 4.31 124.9 3.92 112.3 2.56 9
ID: 3364777 • Letter: 1
Question
1.
Head width (cm)
Biting force (grams/centimer2)
4.31
124.9
3.92
112.3
2.56
93.6
3.72
106.1
4.21
126.6
2.95
108.6
4.74
139.3
2.84
118.0
Imagine that you are examining the functional relationship between head width (measured in centimeters) and biting force (measured in grams per square centimeters) in the tegu lizard.
a)Conduct a linear regression by hand. You will need to show the slope and the intercept of the best-fit line. Also, construct a scatterplot graph with all data-points and trace the best-fit line (if you want you can do the graph in Excel)
b)Conduct a hypothesis test for the significance of the slope of the best fit line (ANOVA table).
c)Calculate a 95% confidence interval on the slope.
d)Calculate the coefficient of determination (r2).
Show your work and explain all results!
Head width (cm)
Biting force (grams/centimer2)
4.31
124.9
3.92
112.3
2.56
93.6
3.72
106.1
4.21
126.6
2.95
108.6
4.74
139.3
2.84
118.0
Explanation / Answer
a. Line of Regression Y on X i.e Y = bo + b1 X
calculation procedure for regression
mean of X = X / n = 3.6563
mean of Y = Y / n = 116.175
(Xi - Mean)^2 = 4.34899
(Yi - Mean)^2 = 1406.44
(Xi-Mean)*(Yi-Mean) = 63.48223
b1 = (Xi-Mean)*(Yi-Mean) / (Xi - Mean)^2
= 63.48223 / 4.34899
= 14.59701
bo = Y / n - b1 * X / n
bo = 116.175 - 14.59701*3.6563 = 62.80397
value of regression equation is, Y = bo + b1 X
Y'=62.804+14.597* X
b.
value of P (.0144) < 0.05 , we reject the null hypothesis
c.
95% CI = [ 4.1047, 25.0893]
d.
calculation procedure for correlation
sum of (x) = x = 29.25
sum of (y) = y = 929.4
sum of (x^2)= x^2 = 111.294
sum of (y^2)= y^2 = 109379.48
sum of (x*y)= x*y = 3461.601
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ x*y - N *(x/N) * (y/N) ]/n-1
= 3461.601 - [ 8 * (29.25/8) * (929.4/8) ]/8- 1
= 7.935
and now to calculate r( x,y) = 7.935/ (SQRT(1/8*3461.601-(1/8*29.25)^2) ) * ( SQRT(1/8*3461.601-(1/8*929.4)^2)
=7.935 / (0.737*13.259)
=0.812
value of correlation is =0.812
coeffcient of determination = r^2 = 0.659
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