Question

A random sample of 1014 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 1014 adults​ surveyed, 537 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.

​(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?

(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

Given number of adults indicated that televisions are a luxury they could do without = 537

Number of adults sampled = 1014

a) point estimate for the population proportion of adults in the country who believe that televisions are a luxury they could do without. = 537/1014 = 0.530

b) np = 0.53 * 1014 = 537

nq = n(1-p) = 0.47*1014 = 477

Since np nq both are greater than or equal to 5 hence we use normal approximation to this binomial distribution problem.

c) 95% CI

level of significance alpha = 0.05

critical value Z_0.025 = 1.96

95% CI = = (0.498 , 0.561)

The lower bound is .498 and the upper bound is .561. We are 95% confident the proportion of adults who believe that televisions are a luxury they could do without is between 0.498 and 0.561.

d) It is possible it possible that a supermajority​ (more than​ 60%) of adults in the country believe that television is a luxury they could do​ without, but it is not​ likely, becuase the 95% confidence interval does not contain .6.

e) proportion adults in the country who believe that televisions are a necessity = 1 - proportion adults in the country who believe that televisions are a luxury.= 1-0.53 = 0.47

level of significance alpha = 0.05

critical value Z_0.025 = 1.96

95% CI =

95% CI = (0.439 , 0.5017)