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) = 21, 25, 23, 24, 20, 15, 25, 21, 17, 24, 26, 22, 18, 19 y (lb) = 125, 125, 120, 125, 130, 120, 145, 130, 130, 130, 130, 140, 110, 115, 130, 130, 130 a. Solve for Σx = Blank 1 b. Solve for Σy = Blank 2 c. Solve for Σx2 = Blank 3 d. Solve for Σy2 = Blank 4 e. Solve for Σxy = Blank 5 f. Solve for Pearson's r = Blank 6 g. Solve for the variance = Blank 7 h. Solve for the slope of the simple linear regression line = Blank 8 i. Solve for the y-intercept of the simple linear regression line = Blank 9 j. (Multiple choice) How would you describe the linear relationship of the data? Write only the letter of your answer: Blank 10 a. Very weak linear relationship b. Weak linear relationship c. Moderate linear relationship d. Strong linear relationship e. Very strong linear relationship
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) = 21, 25, 23, 24, 20, 15, 25, 21, 17, 24, 26, 22, 18, 19 y (lb) = 125, 125, 120, 125, 130, 120, 145, 130, 130, 130, 130, 140, 110, 115, 130, 130, 130
a. Solve for Σx = Blank 1 b. Solve for Σy = Blank 2 c. Solve for Σx2 = Blank 3 d. Solve for Σy2 = Blank 4 e. Solve for Σxy = Blank 5 f. Solve for Pearson's r = Blank 6 g. Solve for the variance = Blank 7 h. Solve for the slope of the simple linear regression line = Blank 8 i. Solve for the y-intercept of the simple linear regression line = Blank 9 j. (Multiple choice) How would you describe the linear relationship of the data? Write only the letter of your answer: Blank 10 a. Very weak linear relationship b. Weak linear relationship c. Moderate linear relationship d. Strong linear relationship e. Very strong linear relationship
Solution
Sure, let's solve these step by step:
a. Σx = sum of all x values = 21 + 25 + 23 + 24 + 20 + 15 + 25 + 21 + 17 + 24 + 26 + 22 + 18 + 19 = 320
b. Σy = sum of all y values = 125 + 125 + 120 + 125 + 130 + 120 + 145 + 130 + 130 + 130 + 130 + 140 + 110 + 115 = 1825
c. Σx2 = sum of squares of all x values = 21^2 + 25^2 + 23^2 + 24^2 + 20^2 + 15^2 + 25^2 + 21^2 + 17^2 + 24^2 + 26^2 + 22^2 + 18^2 + 19^2 = 6810
d. Σy2 = sum of squares of all y values = 125^2 + 125^2 + 120^2 + 125^2 + 130^2 + 120^2 + 145^2 + 130^2 + 130^2 + 130^2 + 130^2 + 140^2 + 110^2 + 115^2 = 238375
e. Σxy = sum of the product of corresponding x and y values = 21125 + 25125 + 23120 + 24125 + 20130 + 15120 + 25145 + 21130 + 17130 + 24130 + 26130 + 22140 + 18110 + 19115 = 41400
f. Pearson's r = (nΣxy - ΣxΣy) / sqrt([nΣx2 - (Σx)^2][nΣy2 - (Σy)^2]) = (1441400 - 3201825) / sqrt([146810 - 320^2][14238375 - 1825^2]) = 0.52
g. Variance = (Σx2/n - (Σx/n)^2) = (6810/14 - (320/14)^2) = 11.84
h. Slope of the simple linear regression line = (nΣxy - ΣxΣy) / (nΣx2 - (Σx)^2) = (1441400 - 3201825) / (146810 - 320^2) = 4.57
i. y-intercept of the simple linear regression line = (Σy/n) - slope*(Σx/n) = (1825/14) - 4.57*(320/14) = 47.14
j. Pearson's r is 0.52, which indicates a moderate linear relationship. So, the answer is c. Moderate linear relationship.
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