Question

1. A pitcher pitched in a game last night and threw 110 pitches, of which 77 pitches were strikes.

(5)        a. Construct a 90% confidence interval for the true proportion of strikes for the entire season (this night’s game should be considered a sample of all games in the season).

(10)      b. At the 99% level test the alternate hypothesis that the true proportion of strikes is greater than 0.67.

(5)        c. What sample size is required if you wanted to be 95% sure that the true proportion of strikes to total pitches is within two percentage points?

Answer 1

a)

p̂ = X / n = 77/110 = 0.7
p̂ ± Z(α/2) √( (p * q) / n)
0.7 ± Z(0.1/2) √( (0.7 * 0.3) / 110)
Z(α/2) = Z(0.1/2) = 1.645
Lower Limit = 0.7 - Z(0.1) √( (0.7 * 0.3) / 110) = 0.628
upper Limit = 0.7 + Z(0.1) √( (0.7 * 0.3) / 110) = 0.772
90% Confidence interval is ( 0.628 , 0.772 )

b)

H0: p <= 0.67

Ha: p > 0.67

Test statistics

z = ( - p) / sqrt [ p ( 1 - p) / n ]

= ( 0.7 - 0.67) / sqrt ( 0.67 ( 1 - 0.67)/ 110 ]

= 0.67

Critical value at 0.01 significance level = 2.326

Since test statistics < 2.326, Fail to reject the null hypothesis

c)

Sample size = Z2/2 * p ( 1 - p) / E2

= 1.962 * 0.7 ( 1 - 0.7) / 0.022

= 2016.84

Sample size = 2017 (Rounded up to nearest integer)