Question

A random sample of 1001 adults in a certain large country was asked​ "Do you pretty much think televisions are a necessity or a luxury you could do​ without?" Of the 1001 adults​ surveyed, 534 indicated that televisions are a luxury they could do without. Complete parts​ (a) through​ (e) below.

a) Obtain a point estimate for the population proportion of adults in the country who believe that televisions are a luxury they could do without.

b) Verify that the requirements for constructing a confidence interval about p are satisfied.

c) Construct and interpret a 95​% confidence interval for the population proportion of adults in the country who believe that televisions are a luxury they could do without. Select the correct choice below and fill in any answer boxes within your choice.

- there is a __% probability the proportion of adults in the country who believe that televisions are a luxury they could do without is between __ and __

- we are __ ​% confident the proportion of adults in the country who believe that televisions are a luxury they could do without is between __ and __

d) Is it possible that a supermajority​ (more than​ 60%) of adults in the country believe that television is a luxury they could do​ without? Is it​ likely?

- It is _____ likely that a supermajority of adults in the country believe that television is a luxury they could do without because the 95​% confidence interval (does not contain or contain) __

e) Use the results of part​ (c) to construct a 95​% confidence interval for the population proportion of adults in the country who believe that televisions are a necessity.

Answer 1

a)

sample success x =	534
sample size          n=	1001
sample proportion (point estimate)= p̂ =x/n=	0.5335

b)

since sample is randomly selected : number of success(534) and number of failures  (467) both are greater than 10, we can use normal approximation of binomial distribution for confidence interval

c)

std error se= √(p*(1-p)/n) =	0.0158
for 95 % CI value of z=	1.960
margin of error E=z*std error   =	0.0309
lower bound=p̂ -E                       =	0.5026
Upper bound=p̂ +E                     =	0.5644

we are 95 ​% confident the proportion of adults in the country who believe that televisions are a luxury they could do without is between 0.5026 and 0.5644

d)

It is not likely that a supermajority of adults in the country believe that television is a luxury they could do without because the 95​% confidence interval does not contain 60%

e)

sample success x =	467
sample size          n=	1001
sample proportion p̂ =x/n=	0.4665
std error se= √(p*(1-p)/n) =	0.0158
for 95 % CI value of z=	1.960
margin of error E=z*std error   =	0.0309
lower bound=p̂ -E                       =	0.4356
Upper bound=p̂ +E                     =	0.4974

from above 95% confidence interval for population proportion =(0.4356,0.4974)