Question

5. An oil company claims that 45% of the people buying gasoline in a particular city buy from their company. A random sample of 400 gasoline purchases in this city is collected. If the company’s claim is correct, find the probability that

a) More than 160, but at most 190 people in the sample purchased gasoline from this company.

b) Fewer than 200 people in the sample purchased gasoline from this company.

c) At least 150 people in the sample purchased gasoline from this company.

Answer 1

n=

400

p=

0.4500

here mean of distribution=μ=np=		180.00
and standard deviation σ=sqrt(np(1-p))=		9.95
for normal distribution z score =(X-μ)/σx

therefore from normal approximation of binomial distribution and continuity correction:

a)

probability =P(160.5<X<190.5)=P((160.5-180)/9.95)<Z<(190.5-180)/9.95)=P(-1.96<Z<1.06)=0.8554-0.025=0.8304

b)

probability =P(X<199.5)=(Z<(199.5-180)/9.95)=P(Z<1.96)=0.9750

c)

probability =P(X>149.5)=P(Z>(149.5-180)/9.95)=P(Z>-3.07)=1-P(Z<-3.07)=1-0.0011=0.9989