Question

A random sample of

10191019

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

10191019

adults​ surveyed,

522522

indicated that televisions are a luxury they could do without. Complete parts​ (a) through​ (e) below.Click here to view the standard normal distribution table (page 1).

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Click here to view the standard normal distribution table (page 2).

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​(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.

ModifyingAbove p with caretpequals=nothing

​(Round to three decimal places as​ needed.)

​(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 caretnp

n ModifyingAbove p with caret left parenthesis 1 minus ModifyingAbove p with caret right parenthesisnp1−p

ModifyingAbove p with caret left parenthesis 1 minus ModifyingAbove p with caret right parenthesisp1−p

nn

ModifyingAbove p with caretp

is

nothing​,

which is

▼

greater than or equal to

less than

​10, and the

▼

sample size

population proportion

sample proportion

population size

▼

is stated to not be

cannot be assumed to be

can be assumed to be

is stated to be

less than or equal to​ 5% of the

▼

sample size.

population size.

sample proportion.

population proportion.

​(Round to three decimal places as​ needed.)

​(c) Construct and interpret a

9595​%

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.We are

nothing​%

confident the proportion of adults in the country who believe that televisions are a luxury they could do without is between

nothing

and

nothing.

B.There is a

nothing​%

chance the proportion of adults in the country who believe that televisions are a luxury they could do without is between

nothing

and

nothing.

​(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

9595​%

confidence interval

▼

does not contain

contains

nothing.

​(Type an integer or a decimal. Do not​ round.)

​(e) Use the results of part​ (c) to construct a

9595​%

confidence interval for the population proportion of adults in the country who believe that televisions are a necessity.The

9595​%

confidence interval is

​(nothing​,nothing​).

​(Round to three decimal places as​ needed.)

Click to select your answer(s).

Please answer all

Answer 1

a)

point estimate for the population proportion =p̂ = x/n = 522/1019 = 0.512

b)

The sample (is stated to be ) a simple random sample, the value of (np^(1-p^)   is 254.597 which is (greater than or equal to)10 and the sample size can be (assumed to be) less than or equal to 5% of (population size).

c)

Level of Significance,   α =    0.05          
Number of Items of Interest,   x =   522          
Sample Size,   n =    1019          
                  
Sample Proportion ,    p̂ = x/n =    0.512          
z -value =   Zα/2 =    1.960   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0157          
margin of error , E = Z*SE =    1.960   *   0.0157   =   0.0307
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.512   -   0.0307   =   0.4816
Interval Upper Limit = p̂ + E =   0.512   +   0.0307   =   0.5430
                  
95%   confidence interval is (   0.482   < p <    0.543   )
                  

We are 95%

confident the proportion of adults in the country who believe that televisions are a luxury they could do without is between 0.482 and 0.543

d)

it is (possible but 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 0.60)

e)

Level of Significance,   α =    0.05          
Number of Items of Interest,   x =   497          
Sample Size,   n =    1019          
                  
Sample Proportion ,    p̂ = x/n =    0.488          
z -value =   Zα/2 =    1.960   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0157          
margin of error , E = Z*SE =    1.960   *   0.0157   =   0.0307
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.488   -   0.0307   =   0.4570
Interval Upper Limit = p̂ + E =   0.488   +   0.0307   =   0.5184
                  
95%   confidence interval is (   0.457   < p <    0.518   )