Question

Below are the average heights for American boys in 1990. Age (years) birth 2 3 Height (cm) 50.8 83.8 91.4 5 106.6 119.3 137.1 157.5

Answer 1

Answer:

The regression equation is defined as,

Y = a +bX

Where a = intercept, and b= slope of the regression equation

The least square estimate of intercept and slope are,

ΣΥΣΧ? - ΣΧΣΧΥ η (ΣΧ2) - (Σχ)2

και η (ΣΧΥ) - ΣxΣΥ η (Σ x2) - (Σ x)?

Form the data values, the values are calculated as,

	X	Y	X^2	Y^2	XY
	0	50.8	0	2580.64	0
	2	83.8	4	7022.44	167.6
	3	91.4	9	8353.96	274.2
	5	106.6	25	11363.56	533
	7	119.3	49	14232.49	835.1
	10	137.1	100	18796.41	1371
	14	157.5	196	24806.25	2205
Sum	41	746.5	383	87155.75	5385.9

ΣΥΣΧ2 - ΣΧΣΧΥ η (ΣΧ2) - (ΣΧ)2 746.5 x 383 – 41 x 5385.9 - = 65.0876 7 (383) – (41)?

Η ΧΥ) ΣΧΣΥ η(Σ Χ2) - (Σ x)? 7 (5385.9) - 41 x 746.5 - 7.0948 7 (383) – (41)

Y = a + bX = 65.0876 +7.0948 X

The correlation coefficient is obtained using the formula

[(A) - (αλζ) «] [(x) - (X) «1 AX-Xu ΤΕ

7 = 7 (5385.9) - 41 x 746.5 == 0.976131 [7(383) – (41)2] [7 (87155.75) – (746.5)?]

The critical value for correlation coefficient is obtained using the critical value table,

re= 0.666 → For degree of freedom, v=1-2=7-2=5 and a = 0.05

r=0.976131 > re = 0.666 Significant

Part (f)

The least square estimated regression equation is,

Y = a +bX = 65.0876 + 7.0948 X

For X =64, estimated height is Yhat = 65.0876 + 7.0948 *64= 519.1548

Foe X = 64, the answer is not reasonable because 64 is outside the the domain of the least square estimation.

Part(I)

Slope, b= 7.0948 7.095

As age increases by one year, the average height increase by 7.095 centimeters.

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