Question

In studies for a medication, 6% of patients gained weight as a side effect. Suppose 703 patients are randomly selected. Use the normal approximation to the binomial to approximate the probability that:

A. Exactly 43 patients will gain weight as a side effect

B. No more than 43 patients will gain weight as a side effect

C. At least 57 patients will gain weight as a side effect. What does this result suggest?

Answer 1

P(gained weight as a side effect), p = 0.06

q = 1 - p = 0.94

Sample size, n = 703

P(X < A) = P(Z < (A - mean)/standard deviation)

Mean = np

= 703 x 0.06

= 42.18

Standard deviation =

=

= 6.2968

A. P(exactly 43 patients will gain weight as side effect) = P(X = 43)

= P(42.5 < X < 43.5)

= P(X < 43.5) - P(X < 42.5)

= P(Z < (43.5 - 42.18)/6.2968) - P(Z < (42.5 - 42.18)/6.2968)

= P(Z < 0.21) - P(Z < 0.05)

= 0.5832 - 0.5199

= 0.0633

B. P(no more than 43 patients will gain weight as a side effect) = P(X 43)

= P(X < 43.5) {with continuity correction)

= 0.5832

C. P(at least 57 patients will gain weight as a side effect) = P(X 57)

= 1 - P(X < 56.5)

= 1 - P(Z < (56.5 - 42.18)/6.2968)

= 1 - P(Z < 2.27)

= 1 - 0.9884

= 0.0116

Probability that at least 57 patients will gain weight as a side effect is less than 0.05. SO, the event is unusual. We are not likely to get at least 57 patients who will gain weight as a side effect.