In a random survey of 204 self-employed individuals, 16 reported having had thei

Question

In a random survey of

204

self-employed individuals,

16

reported having had their tax returns audited by the IRS in the past year. Estimate the proportion of self-employed individuals nationwide who've been audited by the IRS in the past year.

a) Check the assumptions and conditions for constructing a confidence interval.

b) Construct a

98%

confidence interval.

c) Interpret your interval.

d) Explain what

"98%

confidence" means in this context.

a) Check the assumptions and conditions. Choose the correct answer below.

A.

The independence assumption is violated; whether one person is "less likely" to have his or her tax returns audited by the IRS is dependent on whether another person is "less likely" to have his or her tax returns audited by the IRS.

B.

This was a random sample of less than 10% of all self-employed taxpayers; there are less than 10 successes, so the sample is not large enough.

C.

This was a random sample of less than 10% of all self-employed taxpayers; there were at least 10 successes and 10 failures, so the sample is large enough. The data appear to be independent.

D.

This was a random sample of more than 10% of all self-employed taxpayers; the sample is too large.

b) Select the correct choice below and, if necessary, fill in the answer boxes within your choice.

A.

The

98%

confidence interval is

left parenthesis nothing % comma nothing % right parenthesis%,%.

(Round to one decimal place as needed.)

B.

The conditions for constructing a confidence interval are not satisfied.

c) Interpret your interval. Select the correct answer below.

A.

There is

98%

confidence that of all individuals who had their tax returns audited in the past year, the proportion of self-employed individuals falls within the confidence interval bounds.

B.

There is

9898%

confidence that the proportion of self-employed individuals who had their tax returns audited in the past year falls within the confidence interval bounds.

C.

There is

98%

confidence that the proportion of all individuals who had their tax returns audited in the past year falls within the confidence interval bounds.

D.

The conditions for creating a confidence interval are not satisfied.

d) Explain what

"9898%

confidence" means in this context. Select the correct answer below.

A.

If we were to select a new sample of self-employed individuals, there is a

9898%

chance that the true proportion of all self-employed individuals who were audited will change.

B.

If we were to select repeated samples of

204

self-employed individuals, we'd expect about

98%

of the confidence intervals we

Explanation / Answer

a) option C is correct

This was a random sample of less than 10% of all self-employed taxpayers; there were at least 10 successes and 10 failures, so the sample is large enough. The data appear to be independent.

b)

The 98% confidence interval is 3.5 % ; 12.2%

c) option B is correct

d)

option B is correct

                                                                                           x                    =   16 sample size n                    = 204 sample proportion     p x/n= 0.0784 std error       =Se            =(p*(1-p)/n) = 0.0188 for 98 % CI value of z= 2.3263 margin of error E=z*std error                            = 0.0438 lower confidence bound=sample proportion-margin of error 0.035 Upper confidence bound=sample proportion+margin of error 0.122

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