Question

Let x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of n = 6 professional basketball players gave the following information.

x 67 64 75 86 73 73

y 42 39 48 51 44 51

(a) Verify that Σx = 438, Σy = 275, Σx2 = 32264, Σy2 = 12727, Σxy = 20231, and r ≈ 0.827

(b) Use a 5% level of significance to test the claim that ρ > 0. (Round your answers to two decimal places.)

t

critical t

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 insufficient evidence that ρ > 0.

Fail to reject the null hypothesis, there is sufficient evidence that ρ > 0.

(c) Verify that Se ≈ 3.1191, a ≈ 6.564, b ≈ 0.5379, and x ≈ 73.000.

Se

a

b

x bar

(d) Find the predicted percentage y hat of successful field goals for a player with x = 73% successful free throws. (Round your answer to two decimal places.)

% =

(e) Find a 90% confidence interval for y when x = 73. (Round your answers to one decimal place.)

lower limit % =

upper limit % =

(f) Use a 5% level of significance to test the claim that β > 0. (Round your answers to two decimal places.)

t

critical t

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 insufficient evidence that β > 0.

Fail to reject the null hypothesis, there is sufficient evidence that β > 0.

(g) Find a 90% confidence interval for β. (Round your answers to three decimal places.)

lower limit

upper limit

Interpret its meaning.

For every percentage increase in successful free throws, the percentage of successful field goals increases by an amount that falls within the confidence interval.

For every percentage increase in successful free throws, the percentage of successful field goals decreases by an amount that falls outside the confidence interval.

For every percentage increase in successful free throws, the percentage of successful field goals decreases by an amount that falls within the confidence interval.

For every percentage increase in successful free throws, the percentage of successful field goals increases by an amount that falls outside the confidence interval.

Thank you in advance!

Answer 1

a)'

ΣX =	438.000
ΣY=	275.000
ΣX2 =	32264.000
ΣY2 =	12727.000
ΣXY =	20231.000
r =	0.827

b)

test statistic t =		r*(√(n-2)/(1-r2))=		2.94
t crit =				2.13

Reject the null hypothesis, there is sufficient evidence that ρ > 0

c)

Se =√(SSE/(n-2))=				3.11910
a=				6.564
b=				0.5379
X̅=				73.0000

d)

predicted value =

45.83

e)

std error of confidence interval =				s*√(1+1/n+(x0-x̅)2/Sxx)=			3.3690
for 90 % confidence and 4degree of freedom critical t=							2.132
lower limit =		38.7
uppr limit =		53.0

f)

test statistic t =		r*(√(n-2)/(1-r2))=		2.94
t crit =				2.13

Reject the null hypothesis, there is sufficient evidence that β > 0.

g)

std error of slope sb1 =				s/√SSx=		0.1832
for 90 % confidence and -2degree of freedom critical t=						2.1320
90% confidence interval =b1 -/+ t*standard error=					(0.147,0.928)
lower limit =		0.147
uppr limit =		0.928

For every percentage increase in successful free throws, the percentage of successful field goals increases by an amount that falls within the confidence interval.