Question

A fair coin is tossed n = 80 times. Let p̂ be the sample proportion of heads. Find P(0.41 < p̂ < 0.65).
(Round your answer to four decimal places.)

Answer 1

Here, μ = 0.5, σ = 0.0559, x1 = 0.41 and x2 = 0.65. We need to compute P(0.41<= X <= 0.65). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.41 - 0.5)/0.0559 = -1.61
z2 = (0.65 - 0.5)/0.0559 = 2.68

Therefore, we get
P(0.41 <= X <= 0.65) = P((0.65 - 0.5)/0.0559) <= z <= (0.65 - 0.5)/0.0559)
= P(-1.61 <= z <= 2.68) = P(z <= 2.68) - P(z <= -1.61)
= 0.9963 - 0.0537
= 0.9426