Question

In studies for a​ medication, 4 percent of patients gained weight as a side effect. Suppose 749 patients are randomly selected. Use the normal approximation to the binomial to approximate the probability that ​(a) exactly 25 patients will gain weight as a side effect. ​(b) 25 or fewer patients will gain weight as a side effect. ​(c) 18 or more patients will gain weight as a side effect. ​(d) between 25 and 37​, ​inclusive, will gain weight as a side effect. ​(a) ​P(Xequals25​)equals nothing ​(Round to four decimal places as​ needed) ​(b) ​P(Xless than or equals25​)equals nothing ​(Round to four decimal places as​ needed) ​(c) ​P(Xgreater than or equals18​)equals nothing ​(Round to four decimal places as​ needed) ​(d) ​P(25less than or equalsXless than or equals37​)

Answer 1

Solution :

Given that n = 749 , p = 0.04

=> q = 1 - p = 0.96

=> Mean μ = n*p

= 749*0.04

= 29.96

=> Standard deviation σ = sqrt(n*p*q)

= sqrt(749*0.04*0.96)

= 5.3630

(a)
=> P(X = 25) = P(24.5 < X < 25.5)

= P((24.5 - 29.96)/5.3630 < (X - μ)/σ < (25.5 - 29.96)/5.3630)

= P(-1.0181 < Z < -0.8316)

= P(Z < -0.8316) - P(Z < -1.0181)

= 0.2033 - 0.1539

= 0.0494

(b)
=> P(X <= 25) = P((X - μ)/σ <= (25 - 29.96)/5.3630)

= P(Z <= -0.9249)

= 1 - P(Z < 0.9249)

= 1 - 0.8212

= 0.1788

(c)
=> P(X >= 18) = P((X - μ)/σ >= (18 - 29.96)/5.3630)

= P(Z >= -2.2301)

= P(Z < 2.2301)

= 0.9871

(d)
=> P(25 <= X <= 37) = P((25 - 29.96)/5.3630 < (X - μ)/σ < (37 - 29.96)/5.3630)

= P(-0.9249 < Z < 1.3127)

= P(Z < 1.3127) - P(Z < -0.9249)

= 0.9049 - 0.1788

= 0.7261