Question

On Monday SF Giants pitcher Matt Moore pitched 8 innings and threw 93 pitches of which 62 pitches were strikes.

a. Construct a 99% confidence interval for the true proportion(?) of strikes.

b. At the 10% level test the null hypothesis that the true proportion of strikes is greater than 0.60.

c. What sample size is required if you wanted to be 95% sure that the true proportion of strikes to total pitches is within 1 percentage point.

Answer 1

Solution

a.  Construct a 99% confidence interval for the true proportion() of ?strikes.Proportion of strikes

(P) = 62/93= 0.6667, q= 1-P= 1-0.6667=0.3333P-ZP*q)/n]0.5<P+ZP*q)/n]^0.599% confidence interval= 0.99/2=0.495= z= 2.576This implies that 0.6667-{2.576[(0.6667*0.333)/93] ^0.5= 0.6667-0.1258=0.54080.6667+ {2.576[(0.6667*0.333)/93] ^0.5= 0.7926We are 99% confident that the true proportion of strikes is between 54.08% and 79.26%

b.  At the 10% level test the null hypothesis that the true proportion of strikes is greater than 0.60

. H0: P?0.6H1: P>0.6Z= (0.6667-0.6)/ [(0.6*0.4)/93)]^0.5= 1.31= 0.4049Area= 0.5+ 0.4049=0.9049P-value=1- 0.9049=0.0951Since the alpha level of 10 %( 0.1) is larger than the P-value of 0.0951 we reject thenull hypothesis that the true proportion of strikes is less than or equal to 0.60

c.  What sample size is required if you wanted to be 95% sure that the true proportion of strikes to total pitches is within 1 percentage point

Sample size= (Z2Pq)/e2 = (1.96^2*0.6667*0.333)/0.01^2= 8536.46= 8537 pitches

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