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	26	22	26	20	15	25	21	17	24	26	22	18	19
y (lb)	129	128	118	123	130	120	145	130	130	130	130	140	110	115

In this setting we have Σx = 301, Σy = 1778, Σx² = 6637, Σy² = 226,928, and Σxy = 38,412.

(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 four decimal places.)

x	=
y	=
b	=
ŷ	=	+  x

(b) Draw a scatter diagram displaying the data. Graph the least-squares line on your scatter diagram. Be sure to plot the point (x, y).

(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) Test the claim that the population correlation coefficient ρ is positive at the 1% level of significance. (Round your test statistic to three decimal places.)

t =

Find or estimate the P-value of the test statistic.

P-value > 0.2500.125 < P-value < 0.250    0.100 < P-value < 0.1250.075 < P-value < 0.1000.050 < P-value < 0.0750.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.0100.0005 < P-value < 0.005P-value < 0.0005

Conclusion

Reject the null hypothesis. There is sufficient evidence that ρ > 0.Reject the null hypothesis. There is insufficient evidence that ρ > 0.    Fail to reject the null hypothesis. There is sufficient evidence that ρ > 0.Fail to reject the null hypothesis. There is insufficient evidence that ρ > 0.

(e) If a female baby weighs 22 pounds at 1 year, what do you predict she will weigh at 30 years of age? (Round your answer to two decimal places.)
lb

(f) Find S_e. (Round your answer to two decimal places.)
S_e =

(g) Find a 95% confidence interval for weight at age 30 of a female who weighed 22 pounds at 1 year of age. (Round your answers to two decimal places.)

lower limit	lb
upper limit	lb

(h) Test the claim that the slope β of the population least-squares line is positive at the 1% level of significance. (Round your test statistic to three decimal places.)

t =

Find or estimate the P-value of the test statistic.

P-value > 0.2500.125 < P-value < 0.250    0.100 < P-value < 0.1250.075 < P-value < 0.1000.050 < P-value < 0.0750.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.0100.0005 < P-value < 0.005P-value < 0.0005

Conclusion

Reject the null hypothesis. There is sufficient evidence that β > 0.Reject the null hypothesis. There is insufficient evidence that β > 0.    Fail to reject the null hypothesis. There is sufficient evidence that β > 0.Fail to reject the null hypothesis. There is insufficient evidence that β > 0.

(i) Find an 80% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)

lower limit
upper limit

Interpretation

For each pound less a female infant weighs at 1 year, the adult weight increases by an amount that falls within the confidence interval.For each pound more a female infant weighs at 1 year, the adult weight increases by an amount that falls within the confidence interval.    For each pound less a female infant weighs at 1 year, the adult weight increases by an amount that falls outside the confidence interval.For each pound more a female infant weighs at 1 year, the adult weight increases by an amount that falls outside the confidence interval.

Answer 1

From the given data,

Σx = 301, Σy = 1778, Σx2 = 6637, Σy2 = 226,928, and Σxy = 38,412.

a)

b)

c)

r =0.4293
r2=0.1843

18.43% variation in y is explained by the least squares model.

d) Test statistic and p-value for test of slope and correlation are same.

t=1.647 , df=12

The p-value is 0.063.

p-value=0.063/2=0.0315 0.025 < P-value < 0.0500

Therefore, Since p-value is greater than 0.01.

Fail to reject the null hypothesis. There is insufficient evidence that ρ > 0.

e) y=102.967+1.118*22

y=127.563=127.56

f) Se=6.76