Question

how strongly do physical characteristics of sisters and brothers correlate? the data in the table give the heights (in inches) of 12 adult pairs.
brother\tsister
71\t69
68\t64
66\t65
67\t63
70\t65
71\t62
70\t65
73\t64
72\t66
65\t59
66\t62
70\t64
to access the data, click the link for your preferred software format.
csv excel (xls) excel (xlsx) jmp mac-text minitab14-18 minitab18+ pc-text r spss ti crunchit!
assume damien is 70 inches tall. predict the height of his sister tonya. give your answer to one decimal place.
tonyas predicted height:
in

Explanation:

Step1: Calculate mean of Brother's height

First, we sum up all the Brother's heights: \(71 + 68 + 66 + 67 + 70 + 71 + 70 + 73 + 72 + 65 + 66 + 70\)
\(= 71\times2 + 68 + 66\times2 + 67 + 70\times3 + 73 + 72 + 65\)
\(= 142 + 68 + 132 + 67 + 210 + 73 + 72 + 65\)
\(= (142 + 68) + (132 + 67) + (210 + 73) + (72 + 65)\)
\(= 210 + 199 + 283 + 137\)
\(= 210 + 199 = 409; 283 + 137 = 420; 409 + 420 = 829\)
Mean of Brother's height (\(\bar{x}\)) = \(\frac{829}{12} \approx 69.083\)

Step2: Calculate mean of Sister's height

Sum of Sister's heights: \(69 + 64 + 65 + 63 + 65 + 62 + 65 + 64 + 66 + 59 + 62 + 64\)
\(= 69 + (64\times3) + (65\times3) + 63 + 62\times2 + 66 + 59\)
\(= 69 + 192 + 195 + 63 + 124 + 66 + 59\)
\(= 69 + 192 = 261; 195 + 63 = 258; 124 + 66 = 190; 261 + 258 = 519; 190 + 59 = 249; 519 + 249 = 768\)
Mean of Sister's height (\(\bar{y}\)) = \(\frac{768}{12} = 64\)

Step3: Calculate slope (\(b_1\)) and intercept (\(b_0\)) of regression line

First, calculate the numerator and denominator for slope.
Let \(x_i\) be Brother's height, \(y_i\) be Sister's height.
Numerator (\(S_{xy}\)) = \(\sum (x_i - \bar{x})(y_i - \bar{y})\)
Denominator (\(S_{xx}\)) = \(\sum (x_i - \bar{x})^2\)

We can calculate \(S_{xy}\) and \(S_{xx}\) using the data:

For each pair:
1. (71,69): \(x - \bar{x} = 71 - 69.083 = 1.917; y - \bar{y} = 69 - 64 = 5; (x - \bar{x})(y - \bar{y}) = 1.917\times5 = 9.585; (x - \bar{x})^2 = 1.917^2 \approx 3.675\)
2. (68,64): \(x - \bar{x} = 68 - 69.083 = -1.083; y - \bar{y} = 0; (x - \bar{x})(y - \bar{y}) = 0; (x - \bar{x})^2 = 1.173\)
3. (66,65): \(x - \bar{x} = 66 - 69.083 = -3.083; y - \bar{y} = 1; (x - \bar{x})(y - \bar{y}) = -3.083; (x - \bar{x})^2 = 9.505\)
4. (67,63): \(x - \bar{x} = 67 - 69.083 = -2.083; y - \bar{y} = -1; (x - \bar{x})(y - \bar{y}) = 2.083; (x - \bar{x})^2 = 4.339\)
5. (70,65): \(x - \bar{x} = 70 - 69.083 = 0.917; y - \bar{y} = 1; (x - \bar{x})(y - \bar{y}) = 0.917; (x - \bar{x})^2 = 0.841\)
6. (71,62): \(x - \bar{x} = 1.917; y - \bar{y} = -2; (x - \bar{x})(y - \bar{y}) = -3.834; (x - \bar{x})^2 = 3.675\)
7. (70,65): \(x - \bar{x} = 0.917; y - \bar{y} = 1; (x - \bar{x})(y - \bar{y}) = 0.917; (x - \bar{x})^2 = 0.841\)
8. (73,64): \(x - \bar{x} = 73 - 69.083 = 3.917; y - \bar{y} = 0; (x - \bar{x})(y - \bar{y}) = 0; (x - \bar{x})^2 = 15.343\)
9. (72,66): \(x - \bar{x} = 72 - 69.083 = 2.917; y - \bar{y} = 2; (x - \bar{x})(y - \bar{y}) = 5.834; (x - \bar{x})^2 = 8.509\)
10. (65,59): \(x - \bar{x} = 65 - 69.083 = -4.083; y - \bar{y} = -5; (x - \bar{x})(y - \bar{y}) = 20.415; (x - \bar{x})^2 = 16.671\)
11. (66,62): \(x - \bar{x} = -3.083; y - \bar{y} = -2; (x - \bar{x})(y - \bar{y}) = 6.166; (x - \bar{x})^2 = 9.505\)
12. (70,64): \(x - \bar{x} = 0.917; y - \bar{y} = 0; (x - \bar{x})(y - \bar{y}) = 0; (x - \bar{x})^2 = 0.841\)

Now sum up the \((x - \bar{x})(y - \bar{y})\) terms:
\(9.585 + 0 - 3.083 + 2.083 + 0.917 - 3.834 + 0.917 + 0 + 5.834 + 20.415 + 6.166 + 0\)
\(= (9.585 - 3.083) + (2.083 + 0.917) + (-3.834 + 0.917) + (5.834 + 20.415) + 6.166\)
\(= 6.502 + 3 + (-2.917) + 26.249 + 6.166\)
\(= 6.502 + 3 = 9.502; -2.917 + 26.249 = 23.332; 9.502 + 23.332 = 32.834; 32.834 + 6.166 = 39\)
So \(S_{xy} = 39\)

Sum up the \((x - \bar{x})^2\) terms:
\(3.675 + 1.173 + 9.505 + 4.339 + 0.841 + 3.675 + 0.841 + 15.343 + 8.509 + 16.671 + 9.505 + 0.841\)
\(= (3.675 + 1.173) + (9.505 + 4.339) + (0.841 + 3.675) + (0.841 + 15.343) + (8.509 + 16.671) + (9.505 + 0.841)\)
\(= 4.848 + 13.844 + 4.516 + 16.184 + 25.18 + 10.346\)
\(= 4.848 + 13.844 = 18.692; 4.516 + 16.184 = 20.7; 25.18 + 10.346 = 35.526; 18.692 + 20.7 = 39.392; 39.392 + 35.526 = 74.918\)
So \(S_{xx} \approx 74.918\)

Slope \(b_1 = \frac{S_{xy}}{S_{xx}} = \frac{39}{74.918} \approx 0.5206\)

Intercept \(b_0 = \bar{y} - b_1\bar{x} = 64 - 0.5206\times69.083 \approx 64 - 35.97 \approx 28.03\)

Step4: Predict Tonya's height

Regression equation: \(y = b_0 + b_1x\)
Given \(x = 70\) (Damien's height)
\(y = 28.03 + 0.5206\times70 \approx 28.03 + 36.442 \approx 64.472 \approx 64.5\) (rounded to one decimal place)

Answer:

\(64.5\)