Question

Suppose that a new treatment is successful in curing a common ailment

60%

of the time. If the treatment is tried on a random sample of

55

patients, approximate the probability that at most

31

will be cured. Use the normal approximation to the binomial with a correction for continuity.

Round your answer to at least three decimal places. Do not round any intermediate steps.

(If necessary, consult a list of formulas.)

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 55 * 0.6 ) = 33
Variance = n * P * Q = ( 55 * 0.6 * 0.4 ) = 13.2
Standard deviation = √(variance) = √(13.2) = 3.6332

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

P ( X <= 31 ) = ?
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 31 + 0.5 ) = P ( X < 31.5 )

Standardizing the value
Z = ( X - µ ) / σ
Z = ( 31.5 - 33 ) / 3.6332
Z = -0.41
P ( ( X - µ ) / σ ) < ( 31.5 - 33 ) / 3.6332 )
P ( X < 31.5 ) = P ( Z < -0.41 )
P ( X < 31.5 ) = 0.3409