A nationwide study in 2003 indicated that 95 of college stud

A nationwide study in 2003 indicated that 95% of college students with cell phones send and receive text messages with their phones. Suppose a simple random sample of n=1136 of college students with cell phones is obtained.

(a). Let P_hat denote the sample proportion of college students with cell phones who send and receive text messages. Find the following.

(i). E(P_hat)

(ii). SD(P_hat)

(b). What is the sampling distribution model for P_hat?(Check with the rule of thumb np?10 and n(1-p)?10.)

(C).What is the probability that 1065 or fewer college students in the sample send and receive text messages with their cell phones?

(d).If you want to reduce SD(P_hat) by half, how much larger does the sample size n needs to be?

Solution

i) E(P_hat) = np=1136 x 0.95=1079.2

ii)SD(P_hat)= sqrt(npq)=sqrt(1136 x 0.95 x (1-0.95)) =7.3457

iii) Sampling distribution model= Normal distribution

iv)Probability that 1065 or fewer college students in the sample send and receive text messages with their cell phones =P(x<1065) =P(z(=1065-1079.2)/7.3457)<-1.93)

from Normal distribution area table P(z<-1.93)= 0.5-0.4732=0.0268

v) if SD(P_hat) is reduced to half i.e 7.3457/2=3.6785

sqrt(npq)=sqrt(n x 0.95 x (1-0.95)) =3.6785

   nx0.95x0.05 =3.6785²

n=285