Question

Suppose we toss a coin 100 times and get 48 heads. Clearly pˆ = X¯ = 0.48

A) Derive the (large sample) confidence interval for p, assuming the confidence level is 90%

B) Test whether it is a fair coin assuming the significance level α = 0.05.

Please write down the null hypothesis, the alternative hypothesis and the test statistics.

Answer 1

Part a)

p̂ = 48 / 100 = 0.48
p̂ = 1 - p̂ = 0.52
n = 100
p̂ ± Z(α/2) √( (p * q) / n)
0.48 ± Z(0.1/2) √( (0.48 * 0.52) / 100)
Z(α/2) = Z(0.1/2) = 1.645
Lower Limit = 0.48 - Z(0.1) √( (0.48 * 0.52) / 100) = 0.3978
upper Limit = 0.48 + Z(0.1) √( (0.48 * 0.52) / 100) = 0.5622
90% Confidence interval is ( 0.3978 , 0.5622 )
( 0.3978 < P < 0.5622 )

To Test :-

H0 :- P = 0.5 ( A coin is fair )
H1 :- P ≠ 0.5 ( A coin is biased )

P0 = 0.5
q0 = 1 - P0 = 0.5
n = 100
P = X / n = 48/100 = 0.48

Test Statistic :-
Z = ( P - P0 ) / ( √((P0 * q0)/n)
Z = ( 0.48 - 0.5 ) / ( √(( 0.5 * 0.5) /100))
Z = -0.4

Test Criteria :-
Reject null hypothesis if Z < -Z(α/2)
Z(α/2) = Z(0.05/2) = 1.96
Z > -Z(α/2) = -0.4 > -1.96, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Decision based on P value
P value = 2 * P ( Z > -0.4 )
P value = 0.6892
Reject null hypothesis if P value < α = 0.05
Since P value = 0.6892 > 0.05, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

There is sufficient evidence to support the claim that the coin in fair.