A statistics student wants to estimate the probability p tha

A statistics student wants to estimate the probability p that a certain coin will land on heads when it is flipped. The student flips the coin independently 400 times, obtaining 226 heads and 174 tails.

(a) What is a 95% confidence interval for the actual probability of heads?

(b) What is the highest level of confidence among 90%, 95%, 99% with which we can assert that this particular coin is not a fair coin? Show your computations.

Solution

a)
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)=226
Sample Size(n)=400
Sample proportion = x/n =0.565
Confidence Interval = [ 0.565 ±Z a/2 ( Sqrt ( 0.565*0.435) /400)]
= [ 0.565 - 1.96* Sqrt(0.001) , 0.565 + 1.96* Sqrt(0.001) ]
= [ 0.516,0.614]

b.
Confidence Interval = [ 0.565 ±Z a/2 ( Sqrt ( 0.565*0.435) /400)]
= [ 0.565 - 1.645* Sqrt(0.001) , 0.565 + 1.65* Sqrt(0.001) ]
= [ 0.524,0.606]

with which we can assert that this particular coin is not a fair coin,
as the reason minimum bound is above 0.52