Question

Dorothy Kelly sells life insurance for the Prudence Insurance Company. She sells insurance by making visits to her clients homes. Dorothy believes that the number of sales should depend, to some degree, on the number of visits made. For the past several years, she kept careful records of the number of visits (x) she made each week and the number of people (y) who bought insurance that week. For a random sample of 15 such weeks, the x and y values follow.

x	13	20	17	15	28	5	20	14	22	7	15	29	8	25	16
y	3	13	8	3	8	2	5	6	8	3	5	10	6	10	7

In this setting we have Σx = 254, Σy = 97, Σx² = 5032, Σy² = 763, and Σxy = 1870.

(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 least-squares estimates to four decimal places.)

x	=
y	=
b	=
ŷ	= _______	+ _____x

(b) 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.)

(c) 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 the P value =

(d) In a week during which Dorothy makes 13 visits, how many people do you predict will buy insurance from her? (Round your answer to one decimal place.)

(e) Find S_e. (Round your answer to three decimal places.)

(f) Find a 95% confidence interval for the number of sales Dorothy would make in a week during which she made 13 visits. (Round your answers to one decimal place.)

lower limit	_____sales
upper limit	_____sales

(g) 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 the P value =

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

lower limit _____
upper limit _____

Answer 1

a)

X̅=ΣX/n =	16.93
Y̅=ΣY/n =	6.47

b =0.3112

ŷ =	1.1970+0.3112x

b)

r=	0.722
r2 =	0.522

percentage of variation in y is explained by the least-squares model =52.2%

c)

test statistic t =

r*(√(n-2)/(1-r2))=

3.764

p value =0.001

d)\

predicted value =

5.2

e)

Se =√(SSE/(n-2))=

2.235

f)

std error of confidence interval =				s*√(1+1/n+(x0-x̅)2/Sxx)=			2.3312
for 95 % confidence and 13degree of freedom critical t=							2.160
lower limit =		0.2
uppr limit =		10.3

g)

test statistic t =

r*(√(n-2)/(1-r2))=

3.764

p value =0.001

h)

std error of slope sb1 =				s/√SSx=		0.0827
for 80 % confidence and -2degree of freedom critical t=						1.3500
80% confidence interval =b1 -/+ t*standard error=					(0.2,0.423)
lower limit =		0.200
uppr limit =		0.423