Question

The mean wait time at Social Security Offices is 25 minutes with a standard deviation of 11 minutes. Use this information to answer the following questions:

A.            If you randomly select 40 people what is the probability that their average wait time will be more than 27 minutes?

B.            If you randomly select 75 people what is the probability that their average wait time will be between 23 and 26 minutes?

C.            If you randomly select 100 people what is the probability that their average wait time will be at most 24 minutes?

2.            The proportion of people who wait more than an hour at the Social Security Office is 28%. Use this information to answer the following questions:

A.            If you randomly select 45 people what is the probability that at least 34% of them will wait more than an hour?

B.            If you randomly select 60 people what is the probability that between 25% and 30% of them will wait more than an hour?

C.            If you randomly select 150 people what is the probability that less than 23% of them will wait more than an hour?

Answer 1

1)A)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	25
std deviation   =σ=	11.000
sample size       =n=	40
std error=σx̅=σ/√n=	1.73925

probability =P(X>27)=P(Z>(27-25)/1.739)=P(Z>1.15)=1-P(Z<1.15)=1-0.8749=0.1251

B)

sample size       =n=	75
std error=σx̅=σ/√n=	1.2702

probability =P(23<X<26)=P((23-25)/1.27)<Z<(26-25)/1.27)=P(-1.57<Z<0.79)=0.7852-0.0582=0.7270

C)

sample size       =n=	100
std error=σx̅=σ/√n=	1.1000

probability =P(X<24)=(Z<(24-25)/1.1)=P(Z<-0.91)=0.1814

2)A)

here population proportion=     μp=	0.2800
sample size       =n=	45
std error of proportion=σp=√(p*(1-p)/n)=	0.0669

probability =P(X>0.34)=P(Z>(0.34-0.28)/0.067)=P(Z>0.9)=1-P(Z<0.9)=1-0.8159=0.1841

B)

std error of proportion=σp=√(p*(1-p)/n)=

0.0580

probability =P(0.25<X<0.3)=P((0.25-0.28)/0.058)<Z<(0.3-0.28)/0.058)=P(-0.52<Z<0.35)=0.6368-0.3015=0.3353

C)

std error of proportion=σp=√(p*(1-p)/n)=

0.0367

probability =P(X<0.23)=(Z<(0.23-0.28)/0.037)=P(Z<-1.36)=0.0869