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 44 41 48 51 44 51 (a) Verify that Σx = 438, Σy = 279, Σx2 = 32264, Σy2 = 13059, Σxy = 20493, and r ≈ 0.800. Σx Correct: Your answer is correct. Σy Correct: Your answer is correct. Σx2 Correct: Your answer is correct. Σy2 Correct: Your answer is correct. Σxy Correct: Your answer is correct. r Correct: Your answer is correct. (b) Use a 5% level of significance to test the claim that ρ > 0. (Round your answers to two decimal places.) t Correct: Your answer is correct. critical t Incorrect: Your answer is incorrect. 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 ≈ 2.7729, a ≈ 14.783, b ≈ 0.4345, and x ≈ 73.000. Se Correct: Your answer is correct. a Correct: Your answer is correct. b Correct: Your answer is correct. x Correct: Your answer is correct. (d) Find the predicted percentage y hat of successful field goals for a player with x = 71% successful free throws. (Round your answer to two decimal places.) % (e) Find a 90% confidence interval for y when x = 71. (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 outside the confidence interval. 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.

Answer 1

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 44 41 48 51 44 51

(a) Verify that Σx = 438, Σy = 279, Σx2 = 32264, Σy2 = 13059, Σxy = 20493, and r ≈ 0.800. Σx Correct: Your answer is correct. Σy Correct: Your answer is correct. Σx2 Correct: Your answer is correct. Σy2 Correct: Your answer is correct. Σxy Correct: Your answer is correct. r Correct: Your answer is correct.

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

t =2.67 critical t = 2.13 Conclusion Reject the null hypothesis, there is sufficient evidence that ρ > 0.

(c) Verify that Se ≈ 2.7729, a ≈ 14.783, b ≈ 0.4345, and x ≈ 73.000. Se Correct: Your answer is correct. a Correct: Your answer is correct. b Correct: Your answer is correct. x Correct: Your answer is correct.

(d) Find the predicted percentage y hat of successful field goals for a player with x = 71% successful free throws. = 45.63(Round your answer to two decimal places.) %

(e) Find a 90% confidence interval for y when x = 71. (Round your answers to one decimal place.)

lower limit 43.1 % upper limit 48.1%

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

t = 2.67 critical t =2.13 Conclusion 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 = 0.087 upper limit= 0.782

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.

Regression Analysis
	r²	0.6403	n	6
	r	0.8002	k	1
	Std. Error of Estimate	2.7729	Dep. Var.	y
Regression output						confidence interval
variables		coefficients	std. error	t (df=4)	p-value	90% lower	90% upper
Intercept	a =	14.7828
x	b =	0.4345	0.163	2.668	.0559	0.087	0.782
ANOVA table
Source	SS	df	MS	F	p-value
Regression	54.745	1	54.745	7.12	.0559
Residual	30.755	4	7.689
Total	85.500	5
Predicted values for: y
		90% Confidence Interval		90% Prediction Interval
x	Predicted	lower	upper	lower	upper	Leverage
71	45.63	43.12	48.14	39.21	52.05	0.180