6. Anthropologists use a linear model that relates femur (thigh bone) length to

Question

6. Anthropologists use a linear model that relates femur (thigh bone) length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (i femur) is found. You will find data for the femur length and height (in centimeters) in the table below: ncluding the Femur length (x) | 50. i Height ((y) 48.3 | 45.2 44.7-44.5T427-139.5-T 38.01 178.5 | 173.6 164.8 163.7 168.3 1650lissati 55.8 X 353 'x" | 15,691.82 l a. Calculate the correlation coefficient, r using your formula. Enter yourXY 58,687.63 data into the calculator to ensure you are correct b. Y 1,325.1 c. Test to see if there is a significant relationship between x and y.Use d. Calculate the line of best fit, y'- atbx using the formulas. Check your computations with e. Predict the height of an individual that has a femur length of 47.8 cm. Is the relationship between x and y positively or negatively associated? =.05. your calculator. 2 219,928.63

Explanation / Answer

a. r = [n(xy) - (x)((y)] / sqrt [ (n x2 - (x)2) ( n y2 - (y)2]

r = [8 * 58687.63 - 353 * 1325.1] / sqrt [(8 * 15691.82 - 3532) * (8 * 219928.63 - 1325.12)]

r = 1740.74 /sqrt (925.56 * 3539.03)

r = 1740.74 / 1809.8576 = 0.9618

b. Yes we can say that the relationship is positively associated.

c. Here r = 0.9618

so t = r/ sqrt [(n-2)/(1- r2) ]

t = 0.9618 / sqrt [(8-2)/ (1 - 0.96182)]

t = 8.60725

so here for dF = 8-2 = 6 and alpha = 0.05

tcritical  = 2.4469

so here t > tcritical   so we shall reject the null hypothesis.

d. Here y^ = a + bx

so here n = 8

a = [(y) (x2 ) - (x) (xy)]/ [ n (x2 ) - (x)2 ]

a = [1325.1 * 15691.82 - 353 * 58687.63] / [8 * 15691.82 - 3532]

a = 76497.292/ 925.56 = 82.6497

b = [ n(xy) - (x)((y)]/ [ n (x2 ) - (x)2 ]

b = [8 * 58687.63 - 353 * 1325.1]/ [8 * 15691.82 - 3532]

b = 1740.74/ 925.56 = 1.880742

y^ = 1.8807x + 82.650

e. Here Femur length x = 47.8 cm

y^ = 1.8807 * 47.8 + 82.650 = 172.5492 cm

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