Question

Compute​ P(X) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If​ so, approximate​ P(X) using the normal distribution and compare the result with the exact probability. n=62 p=.5 x=41

Answer 1

Using binomial probability formula,

P(X) = ⁿC_x * p^X * ( 1 - p)^n-X

Now,

np = 62 * 0.5 = 31

n( 1 - p) = 62 * ( 1 - 0.5) = 31

Since np > 5 and n( 1 -p) > 5 , normal approximation is appropriate.

Mean = np = 62 * 0.5 = 31

Standard deviation = sqrt ( np( 1 - p) = sqrt ( 62 * 0.5 * 0.5) = 3.9370

Using normal approximation,

P(X < x) = P(Z < ( x - Mean) / SD)

With continuity correction,

P(X = 41) = P(40.5 < X < 41.5)

= P(X < 41.5) - P(X < 40.5)

= P(Z < ( 41.5 - 31) / 3.9370) - P(Z < ( 40.5 - 31) / 3.9370)

= P(Z < 2.67) - P(Z < 2.41)

= 0.9962 - 0.9920

= 0.0042

Using exact binomial formula,

P(X = 41) = ⁶²C₄₁ * 0.5⁴¹ * 0.5^(62-41)

= 0.0040

Difference between both probabilities is less than 0.05, so the result is approximately similar.