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(π) 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

a)

sample proportion, = 0.6667
sample size, n = 93
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.6667 * (1 - 0.6667)/93) = 0.0489

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.58

Margin of Error, ME = zc * SE
ME = 2.58 * 0.0489
ME = 0.1262

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.6667 - 2.58 * 0.0489 , 0.6667 + 2.58 * 0.0489)
CI = (0.5405 , 0.7929)

b)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.6
Alternative Hypothesis, Ha: p > 0.6

Rejection Region
This is right tailed test, for α = 0.1
Critical value of z is 1.282.
Hence reject H0 if z > 1.282

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.6667 - 0.6)/sqrt(0.6*(1-0.6)/93)
z = 1.31

P-value Approach
P-value = 0.0951
As P-value < 0.1, reject the null hypothesis.

c)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.01

The provided estimate of proportion p is, p = 0.6667
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.6667*(1 - 0.6667)*(1.96/0.01)^2
n = 8536.46

Therefore, the sample size needed to satisfy the condition n >= 8536.46 and it must be an integer number, we conclude that the minimum required sample size is n = 8537
Ans : Sample size, n = 8537 or 8536

if we take pvalue = 0.667 upto 3 decimal answer would be change