Question

A random sample of 1012 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 1012 adults​ surveyed, 514 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.

enter your response here
​(Round to three decimal places as​ needed.)
Part 2
​(b) Verify that the requirements for constructing a confidence interval about p are satisfied.
The sample
▼
is stated to be
can be assumed to be
cannot be assumed to be
is stated to not be
a simple random​ sample, the value of
▼
n
ModifyingAbove p with caret left parenthesis 1 minus ModifyingAbove p with caret right parenthesis
ModifyingAbove p with caret
n ModifyingAbove p with caret
n ModifyingAbove p with caret left parenthesis 1 minus ModifyingAbove p with caret right parenthesis
is

enter your response here​, which is
▼
greater than or equal to
less than
​10, and the
▼
population proportion
sample proportion
population size
sample size

▼
cannot be assumed to be
can be assumed to be
is stated to not be
is stated to be
less than or equal to​ 5% of the
▼
population proportion.
sample size.
sample proportion.
population size.
​(Round to three decimal places as​ needed.)
Part 3
​(c) Construct and interpret a ​% 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.
​(Type integers or decimals rounded to three decimal places as needed. Use ascending​ order.)
A.
There is a

enter your response here​% probability the proportion of adults in the country who believe that televisions are a luxury they could do without is between

enter your response here and

enter your response here.
B.
We are

enter your response here​% confident the proportion of adults in the country who believe that televisions are a luxury they could do without is between

enter your response here and

enter your response here.
Part 4
​(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
▼
possible, but not likely
not possible
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
contains


enter your response here.
​(Type an integer or a decimal. Do not​ round.)
Part 5
​(e) Use the results of part​ (c) to construct a ​% confidence interval for the population proportion of adults in the country who believe that televisions are a necessity.
Lower​ bound:

enter your response here​, Upper​ bound:

enter your response here
​(Round to three decimal places as​ needed.)

Answer 1

Answer: (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.

The point estimate for the population proportion is 514 / 1012 = 0.507.

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

The sample is stated to be a simple random sample.

The value of n * p = 1012 * 0.507 = 514.7796 is greater than or equal to 10.

The value of n * (1 - p) = 1012 * (1 - 0.507) = 497.2204 is greater than or equal to 10.

The population proportion is less than or equal to 5% of the population size.

Therefore, the requirements for constructing a confidence interval 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.

The 95% confidence interval for the population proportion is 0.507 +/- 1.96 * sqrt((0.507 * (1 - 0.507)) / 1012) = 0.507 +/- 0.0245 = (0.4825, 0.5315).

We are 95% confident that the proportion of adults in the country who believe that televisions are a luxury they could do without is between 0.4825 and 0.5315.

(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 possible that a supermajority of adults in the country believe that television is a luxury they could do without because the 95% confidence interval contains 60%. However, it is not likely because the confidence interval is relatively narrow and the point estimate is far from 60%.

(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.

The 95% confidence interval for the population proportion of adults who believe that televisions are a necessity is 1 - 0.5315 = 0.4685 to 1 - 0.4825 = 0.5175. Therefore, the 95% confidence interval for the population proportion of adults who believe that televisions are a necessity is (0.4685, 0.5175).

Explanation: