A character is overpowered if the proportion p of contests w

A character is overpowered if the proportion p of contests won by that character is significantly greater than 0.5.

a) How many contests n must be observed to estimate p with possible 2% error at 90% confidence?

b) Suppose 50 randomly selected contests are observed and the character wins 30 of them. Find an appropriate 90% confidence interval for p.

Solution

a)
Compute Sample Size ( n ) = n=(Z/E)^2*p*(1-p)
Z a/2 at 0.1 is = 1.645
Samle Proportion = 0.5
ME = 0.02
n = ( 1.645 / 0.02 )^2 * 0.5*0.5
= 1691.266 ~ 1692

b)
Confidence Interval For Proportion
CI = p ± Z a/2 Sqrt(p*(1-p)/n)))
x = Mean
n = Sample Size
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x)=30
Sample Size(n)=50
Sample proportion = x/n =0.6
Confidence Interval = [ 0.6 ±Z a/2 ( Sqrt ( 0.6*0.4) /50)]
= [ 0.6 - 1.645* Sqrt(0.005) , 0.6 + 1.65* Sqrt(0.005) ]
= [ 0.486,0.714]