Question

Let x be a random variable that represents blood glucose level after a 12-hour fast. Let y be a random variable representing blood glucose level 1 hour after drinking sugar water (after the 12-hour fast). Units are in milligrams per 10 milliliters (mg/10 ml). A random sample of eight adults gave the following information. Σx = 63.6; Σx2 = 518.66; Σy = 89.9; Σy2 = 1050.19; Σxy = 730.65 x 6.1 8.4 7.0 7.5 8.2 6.8 10.0 9.6 y 9.8 10.4 10.3 11.9 14.2 7.0 14.1 12.2 (a) Find the equation of the least-squares line. (Round your answers to three decimal places.) ŷ = + x (b) Draw a scatter diagram for the data. Graph 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 sample coefficient of determination r2. (Round your answers to three decimal places.) r = r2 = Explain the meaning of r2 in the context of the application. (Round your answer to one decimal place.) % of the variance in blood glucose level ---Select--- is explained by the model and the variance in blood glucose level ---Select--- . (d) If x = 9.0, use the least-squares line to predict y. (Round your answer to two decimal places.) y = Find an 80% confidence interval for your prediction. (Round your answers to two decimal places.) lower limit mg/10 ml upper limit mg/10 ml (e) Use level of significance 1% and test the claim that the population correlation coefficient ρ is not zero. (Round your test statistic to three decimal places.) t = Find or estimate the P-value of the test statistic. P-value > 0.500 0.250 < P-value < 0.500 0.200 < P-value < 0.250 0.150 < P-value < 0.200 0.100 < P-value < 0.150 0.050 < P-value < 0.100 0.020 < P-value < 0.050 0.010 < P-value < 0.020 0.001 < P-value < 0.010 P-value < 0.001 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. Interpret the results. At the 1% level of significance, there is sufficient evidence that there is a linear correlation between the two glucose measurements. At the 1% level of significance, there is sufficient evidence that there is not a linear correlation between the two glucose measurements. At the 1% level of significance, there is insufficient evidence that there is a linear correlation between the two glucose measurements. At the 1% level of significance, there is insufficient evidence that there is not a linear correlation between the two glucose measurements. (f) Find an 85% confidence interval for the slope β of the population-based least-squares line. (Round your answers to three decimal places.) lower limit upper limit Explain its meaning in the context of the application. For each additional mg/10 ml of blood glucose level after a 12-hour fast, the blood glucose level 1 hour after drinking sugar water is expected to decrease by an amount that falls outside the confidence interval. For each additional mg/10 ml of blood glucose level after a 12-hour fast, the blood glucose level 1 hour after drinking sugar water is expected to increase by an amount that falls outside the confidence interval. For each additional mg/10 ml of blood glucose level after a 12-hour fast, the blood glucose level 1 hour after drinking sugar water is expected to decrease by an amount that falls within the confidence interval. For each additional mg/10 ml of blood glucose level after a 12-hour fast, the blood glucose level 1 hour after drinking sugar water is expected to increase by an amount that falls within the confidence interval.

Answer 1

	n=	8.0000
	X̅=ΣX/n	7.95
	Y̅=ΣY/n	11.24
	sx=(√(Σx2-(Σx)2/n)/(n-1))=	1.36
	sy=(√(Σy2-(Σy)2/n)/(n-1))=	2.3886
	Cov=sxy=(ΣXY-(ΣXΣY)/n)/(n-1)=	2.2779
	r=Cov/(Sx*Sy)=	0.6987

a)

y^ =1.516+1.223x

b)

c)

r=Cov/(Sx*Sy)=

0.699

coeff of determination r2 =

0.488

48.8 % of the variance in blood glucose level is explained by the model and the variance in blood glucose level

d)

predicted val=1.516+9*1.223=

12.52

standard error of PI=s*√(1+1/n+(x0-x̅)2/Sxx)=		2.0300
for 80 % CI value of t =	1.440
margin of error E=t*std error   =		2.923
lower confidence bound=xo-E =		9.60
Upper confidence bound=xo+E=		15.44

e)

t=r*(√(n-2)/(1-r2))=

2.392

0.050 < P-value < 0.100

Fail to reject the null hypothesis, there is insufficient evidence that ρ ≠ 0.
At the 1% level of significance, there is insufficient evidence that there is not a linear correlation between the two glucose measurements
f)

std error of slope =se(β1) =s/√Sxx=		0.5111
for 85 % CI value of t =	1.650
margin of error E=t*std error   =		0.843
lower confidence bound=xo-E =		0.379
Upper confidence bound=xo+E=		2.066

For each additional mg/10 ml of blood glucose level after a 12-hour fast, the blood glucose level 1 hour after drinking sugar water is expected to increase by an amount that falls within the confidence interval.