Question

In studies for a​ medication, 9 percent of patients gained weight as a side effect. Suppose 542 patients are randomly selected. Use the normal approximation to the binomial to approximate the probability that ​(a) exactly 42 patients will gain weight as a side effect. ​(b) 42 or fewer patients will gain weight as a side effect. ​(c) 56 or more patients will gain weight as a side effect. ​(d) between 42 and 65​, ​inclusive, will gain weight as a side effect. ​(a) ​P(Xequals42​)equals nothing ​(Round to four decimal places as​ needed)

Answer 1

n = 542, P = 9% = 0.09

Checking the condition for using Normal approximaiton to Binomial

n * P >=5     and n * Q >= 5,   where Q = 1 - P = 1 - 0.09 = 0.91

n * P = 542 * 0.09 = 48.78

n * Q = 542 * 0.91 = 493.22

E ( X ) = n * P = 48.78

Standard deviation ( X ) =

Part a) P ( X = 42 )

Using continuity correction

P ( 42 - 0.5 < X < 42 + 0.5 ) = P ( 41.5 < X < 42.5 )

Standardizing the value

Part b)   P ( X <= 42 )

Using continuity correction

P ( X < 42 + 0.5 ) = P ( X < 42.5 )

Part c)   P ( X >= 56 )

Using continuity correction

P ( X > 56 - 0.5 ) = P ( X > 55.5 )

Part d) P ( 42 < X < 65 )

Using continuity correction

P ( 42 - 0.5 < X < 65 + 0.5 ) = P ( 41.5 < X < 65.5 )