Question

Modern medical practice tells us not to encourage babies to become too fat. Is there a positive correlation between the weight x of a 1-year old baby and the weight y of the mature adult (30 years old)? A random sample of medical files produced the following information for 14 females. x (lb) 20 24 22 25 20 15 25 21 17 24 26 22 18 19 y (lb) 126 124 122 124 130 120 145 130 130 130 130 140 110 115 Σx = 298; Σy = 1,776; Σx2 = 6,486; Σy2 = 226,362; Σxy = 37,990 (a) Find x, y, b, and the equation of the least-squares line. (Round your answers for x and y to two decimal places. Round your answers for least-squares estimates to three decimal places.) x = y = b = ŷ = + x (b) Draw a scatter diagram for the data. Plot the least-squares line on your scatter diagram. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (c) Find the sample correlation coefficient r and the coefficient of determination. (Round your answers to three decimal places.) r = r2 = What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.) % (d) If a female baby weighs 17 pounds at 1 year, what do you predict she will weigh at 30 years of age? (Round your answer to two decimal places.) lb

Answer 1

X	Y	XY	X²	Y²
20	126	2520	400	15876
24	124	2976	576	15376
22	122	2684	484	14884
25	124	3100	625	15376
20	130	2600	400	16900
15	120	1800	225	14400
25	145	3625	625	21025
21	130	2730	441	16900
17	130	2210	289	16900
24	130	3120	576	16900
26	130	3380	676	16900
22	140	3080	484	19600
18	110	1980	324	12100
19	115	2185	361	13225
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
298	1776	37990	6486	226362

Sample size, n =	14
x̅ = Ʃx/n = 298/14 =	21.2857143
y̅ = Ʃy/n = 1776/14 =	126.857143
SSxx = Ʃx² - (Ʃx)²/n = 6486 - (298)²/14 =	142.857143
SSyy = Ʃy² - (Ʃy)²/n = 226362 - (1776)²/14 =	1063.71429
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 37990 - (298)(1776)/14 =	186.571429

a)

x̅ = Ʃx/n = 298/14 = 21.2857143 = 21.29

y̅ = Ʃy/n = 1776/14 = 126.857143 = 126.86

Slope, b = SSxy/SSxx = 186.57143/142.85714 = 1.306

y-intercept, a = y̅ -b* x̅ = 126.85714 - (1.306)*21.28571 = 99.058

Regression equation :

ŷ = 99.058 + (1.306) x

b) Scatter plot:

c)

Correlation coefficient, r = SSxy/√(SSxx*SSyy)

= 186.57143/√(142.85714*1063.71429) = 0.479

Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy)

= (186.57143)²/(142.85714*1063.71429) = 0.229

22.9% variation in y is explained by the least squares model.

d)

Predicted value of y at x = 17

ŷ = 99.058 + (1.306) * 17 = 121.26 lb