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.\[\begin{array}{cccccc}67 & 74 & 56 & 61 & 73 & 71 \\67 & 48 & 52 & 65 & 69 & 50 \\\end{array}\]\[\sum x = 402, \sum y = 351, \sum x^2 = 27264, \sum y^2 = 22321, \sum xy = 20231, \, \text{and} \, r = 0.827.\](a) Verify that \( \bar{x} = 67, \, s_x^2 = 27.5, \, s_x = 5.244, \, \bar{y} = 58.5, \, s_y^2 = 122.7, \, s_y = 11.076, \, \text{and} \, b = 0.827 \).(b) Use a 5% level of significance to test the claim that \( \rho > 0 \). (Round your answers to two decimal places.)\[ t_{\text{critical}} = \_\_\_\_\]Conclusion:\begin{itemize}\item Reject the null hypothesis, there is sufficient evidence that \( \rho > 0 \).\item Fail to reject the null hypothesis, there is insufficient evidence that \( \rho > 0 \).\item Reject the null hypothesis, there is insufficient evidence that \( \rho < 0 \).\item Fail to reject the null hypothesis, there is sufficient evidence that \( \rho < 0 \).\end{itemize}(c) Verify that \( b_1 = -3.1191, \, a = 6.564, \, b_0 = 5.379, \, \text{and} \, \bar{x} = 73.000 \).\[s_e \, \text{(standard error)} = \_\_\_\_\]\( a \) =

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Math

Statistics

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 ficid goals a professional basketball player makes in a season. A random sample of n-6 professional basketball players gave the f 73 73 67 64 75 8 Ly42 30 48 SL 4 S (a) Verity that 2x - 438, y - 275, Ex - 32264, Ey - 12727, Ery - 20231, and r-0.827. Exy (b) Use a 5% level of significance to test the claim that p> 0. (Round your answers to two decimal places.) critical Coodusion O Reject the null hypothesis, there is sufficient evidence that > 0. O Reject the null hypothesis, there is insuficient evidence that a>0. O Fail to reject the null hypothesis, there is insufcient evidence that e>0. O Fail to reject the nul hypothesis, there is sufficient evidence that p> (c) Verity thatS,- 3.1191, a - 6.564, D- 0.5379, and x 73.000. (4) Find the predicted percentage y of successtul field goals for a player with x- 75% successtul free throws. (Round your answer to two dedimal places.) (e) Find a 90% confidence interval for y when x- 75. (Round your answers to one decimal place.) lower limit % upper ime () Use a S% level of significance to test the daim that > 0. (Round your answers to two decimal places.) critical Conclusion O Reject the null hypothesis, there is sufficient evidence that > 0. O Reject the null hypothesis, there is insufficient evidence that s> 0. O Fail to reject the null hypothesis, there is insufficient evidence that > 0. O Fail to reject the nul hypothesis, there is sufficient evidence that >0 (0) Find a 90% contidence interval for . (Round your answers to three decimal places.) lower imit upper imit Interpret ts meaning. O For every percentage increase in successful free throws, the percentage of successful field goals decreases by an amount that falls outside the confidence interval. O For every percentage increase in successful free throws, the percentage of successful field goals decreases by an amount that falls within the confidence interval. O For every percentage increase in successtul free throws, the percentage of successful feld goals increases by an amount that falls outside the confidence interval, O For every percentage increase in successtul free throws, the percentage of successtul field goals increases by an amount that falls within the confidence interval.