Question

1. Baseballs vary somewhat in their rebounding coefficient. A baseball that has a large rebound coefficient will travel further when the same force is applied to it than a ball with a smaller coefficient. To achieve a game in which each batter has an equal opportunity to hit a home run, the balls should have nearly the same rebound coefficient. A standard test has been developed to measure the rebound coefficient of baseballs. A purchaser of large quantities of baseballs requires that the standard deviation σ be less than 2 units. A random sample of 81 baseballs is selected from a large batch of balls and tested. Summary statistics are given as: n = 81, X¯ = 85.296, s = 1.771. a. Test the hypothesis H0 : σ 2 ≥ 2 2 against Ha : σ 2 < 2 2 , where σ 2 is the true variance of the rebound coefficients. b. Construct a 95% confidence interval for the variance of the rebound coefficients. c. What are the assumptions that need to hold in order for the analysis in parts (a) and (b) to be valid?

Answer 1

Part a)

To Test :-

H0 :-

H1 :-  

Test Statistic :-

χ² = ( ( 81-1 ) * 3.1364 ) / 4
χ² = 62.728

Test Criteria :-
Reject null hypothesis if
χ² (1 - 0.05,81 - 1) = 60.391
= 62.728 > 60.391 , hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Decision based on P value
P value = P ( χ² > 62.728 ) = 0.0771
Reject null hypothesis if P value < α = 0.05
Since P value = 0.0771 > 0.05, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

There is insufficient evidence to support the claim that the standard deviation σ be less than 2 units.

Part b)

χ² (0.05/2) = 106.6286
χ² (1 - 0.05/2) ) = 57.1532
Lower Limit = (( 81-1 ) 3.1364 / χ² (0.05/2) ) = 2.3531
Upper Limit = (( 81-1 ) 3.1364 / χ² (0.05/2) ) = 4.3902
95% Confidence interval is ( 2.3531 , 4.3902 )
( 2.3531 < σ² < 4.3902 )

Part c)

Assumption need to hold is that the population from which the sample has been is normal.