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 61 67 75 86 73 73 y 40 39 48 51 44 51 (a) Verify that Σx = 435, Σy = 273, Σx2 = 31889, Σy2 = 12563, Σxy = 19974, and r ≈ 0.814. Σx Σy Σx2 Σy2 Σxy r (b) Use a 5% level of significance to test the claim that ρ > 0. (Use 2 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. Correct: Your answer is correct. (c) Verify that Se ≈ 3.4562, a ≈ 8.064, b ≈ 0.5164, and x ≈ 72.500. Se a b x (d) Find the predicted percentage y hat of successful field goals for a player with x = 71% successful free throws. (Use 2 decimal places.) % (e) Find a 99% confidence interval for y when x = 71. (Use 1 decimal place.) lower limit % upper limit % (f) Use a 5% level of significance to test the claim that β > 0. (Use 2 decimal places.) t critical t

Answer 1

n = 6

x 61 67 75 86 73 73 y 40 39 48 51 44 51

(a) Verify that Σx = 435, Σy = 273, Σx2 = 31889, Σy2 = 12563, Σxy = 19974, and r ≈ 0.814. Σx Σy Σx2 Σy2 Σxy r

x 61 67 75 86 73 73

Σx = 61 + 67 + 75 + 86 + 73 + 73 = 435

y 40 39 48 51 44 51

Σy = 40 + 39 + 48 + 51 + 44 + 51 = 273

  

(b) Use a 5% level of significance to test the claim that ρ > 0. (Use 2 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. Correct: Your answer is correct.

Under Ho: Test statistic is given as:

t critical=

Conclusion: Since t(calculated ) = 2.80 > t(tabulated) i.e. t critical = 2.132 we have sufficient evidence to reject Ho at 5% level of significance. Reject the null hypothesis, there is sufficient evidence that ρ > 0.

(c) Verify that Se ≈ 3.4562, a ≈ 8.064, b ≈ 0.5164, and x ≈ 72.500. Se a b x

x ≈ 72.500

b ≈ 0.5164

a ≈ 8.064

Se ≈ 3.4562

(d) Find the predicted percentage y hat of successful field goals for a player with x = 71% successful free throws. (Use 2 decimal places.) %

the predicted percentage y hat of successful field goals for a player with x = 71% successful free throws is 45.27% . (Use 2 decimal places.)

(e) Find a 99% confidence interval for y when x = 71. (Use 1 decimal place.) lower limit % upper limit %

99% confidence interval for y:

99% confidence interval for y:

Answer: (Use 1 decimal place.) lower limit 28.03% upper limit 65.50%

(f) Use a 5% level of significance to test the claim that β > 0. (Use 2 decimal places.) t critical t

The sample test statistic is:

with df = n-2= 6-2=4

We use the students t- distribution table to find the value the critical region:

tcritical =

Since 3.045(t calculated) >2.132(t tabulated) we have sufficient evidence to reject Ho at 5% level of significance.

Hence we conclude that the slope is positive,