Question

Modern medical practice tells us not to encourage babies to become too fat. Is there a positive correlation between the weight, x, or 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. Please use this data to answer all parts of the question.

x (lbs) 21 25 23 24 20 15 25 21 17 24 26 22 18 19

y (lbs) 125 125 120 125 130 120 145 130 130 130 130 140 110 115

note: For these data, ?̅≈ 21.42, ?? ≈ 3.32, ?̅ ≈ 126.79, ?? ≈ 9.12 1.

a. Find the equation of the least-squares line: ?̂ = ? + ??. Round your results to two decimal places.

b. Compute the coefficient of determination, ? ^2 . What percentage of the variation in ? can be explained by the corresponding variation in ? (be sure to write this statement about weight as a 1-year-old and weight as an adult). What percentage is due to other factors? What does this tell you about how well our model fits the given data?

c. Interpret the slope of your least-squares line in the context of this application (what does the specific numeric value of the slope mean in terms of the relationship between weight as a 1-year-old and weight as an adult?).

Answer 1

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	21	125	441	15625	2625
	25	125	625	15625	3125
	23	120	529	14400	2760
	24	125	576	15625	3000
	20	130	400	16900	2600
	15	120	225	14400	1800
	25	145	625	21025	3625
	21	130	441	16900	2730
	17	130	289	16900	2210
	24	130	576	16900	3120
	26	130	676	16900	3380
	22	140	484	19600	3080
	18	110	324	12100	1980
	19	115	361	13225	2185
Total	300	1775	6572	226125	38220

(a)

The least square regression line is

y' = 1.28*x + 99.25

(b)

The coefficient of determination, ? ^2 is 0.219.

The percentage of the variation in ?, that is weight of the mature adult, can be explained by the corresponding variation in ?, that is the weight of 1-year-old baby, is 21.9%.

The percentage of variation due to other factors is 100% - 21.9% = 78.1%.

Since r-square is less so model is not a good fit.

(c)

For each unit increase is weight of 1-year-old, weight of an adult is increased by 1.28 units.