Question

In: Statistics and Probability

Let x be a random variable that represents the batting average of a professional baseball player....

Let x be a random variable that represents the batting average of a professional baseball player. Let y be a random variable that represents the percentage of strikeouts of a professional baseball player. A random sample of n = 6 professional baseball players gave the following information.

x 0.312 0.278 0.340 0.248 0.367 0.269
y 3.1 8.0 4.0 8.6 3.1 11.1

(a) Verify that Σx = 1.814, Σy = 37.9, Σx2 = 0.558782, Σy2 = 296.39, Σxy = 10.8076, and r ≈ -0.847.

Σx   
Σy
Σx2
Σy2
Σxy
r


(b) Use a 1% level of significance to test the claim that ρ ≠ 0. (Use 2 decimal places.)

t
critical t ±   

Conclusion (Select one)

Reject the null hypothesis, there is sufficient evidence that ρ differs from 0.

Reject the null hypothesis, there is insufficient evidence that ρ differs from 0.     

Fail to reject the null hypothesis, there is insufficient evidence that ρ differs from 0.

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



(c) Verify that Se ≈ 2.0037, a ≈ 25.329, and b ≈ -62.886.

Se   
a
b

(d) Find the predicted percentage  of strikeouts for a player with an x = 0.298 batting average. (Use 2 decimal places.)
%

(e) Find a 90% confidence interval for y when x = 0.298. (Use 2 decimal places.)

lower limit %
upper limit %


(f) Use a 1% level of significance to test the claim that β ≠ 0. (Use 2 decimal places.)

t
critical t ±

Conclusion (Select one)

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

Reject the null hypothesis, there is insufficient evidence that β differs from 0.     

Fail to reject the null hypothesis, there is insufficient evidence that β differs from 0.

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



(g) Find a 90% confidence interval for β and interpret its meaning. (Use 2 decimal places.)

lower limit
upper limit

Interpretation (Select one)

For every unit increase in batting average, the percentage strikeouts increases by an amount that falls within the confidence interval.

For every unit increase in batting average, the percentage strikeouts increases by an amount that falls outside the confidence interval.     

For every unit increase in batting average, the percentage strikeouts decreases by an amount that falls outside the confidence interval.

For every unit increase in batting average, the percentage strikeouts decreases by an amount that falls within the confidence interval.

Solutions

Expert Solution

(a)

Observation x y x^2 y^2 xy
1 0.312 3.1 0.097344 9.61 0.9672
2 0.278 8 0.077284 64 2.224
3 0.34 4 0.1156 16 1.36
4 0.248 8.6 0.061504 73.96 2.1328
5 0.367 3.1 0.134689 9.61 1.1377
6 0.269 11.1 0.072361 123.21 2.9859
SUM 1.814 37.9 0.558782 296.39 10.8076

Observation x y Predicted Y Residuals (e) e^2
1 0.312 3.1 5.70876385 -2.60876385 6.805648823
2 0.278 8 7.846904793 0.153095207 0.023438143
3 0.34 4 3.947941896 0.052058104 0.002710046
4 0.248 8.6 9.733499742 -1.133499742 1.284821666
5 0.367 3.1 2.250006442 0.849993558 0.722489049
6 0.269 11.1 8.412883278 2.687116722 7.22059628
SUM 16.05970401

(e)

Observation x (x-0.30233)^2
1 0.312 0.00009344
2 0.278 0.00059211
3 0.34 0.00141878
4 0.248 0.00295211
5 0.367 0.00418178
6 0.269 0.00111111
SUM 1.814 0.010349333
AVERAGE 0.3023333


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