An interactive poll found that 331 of 2,297adults aged 18 or older have at least

Question

An interactive poll found that 331 of 2,297adults aged 18 or older have at least one tattoo.

(1) Construct a 90% confidence interval for the proportion of adults with at least one tattoo.

(2) Construct a 99% confidence interval for the proportion of adults with at least one tattoo.

(3) What is the effect of increasing the level of confidence on the width of the interval?

(1) Construct the 90% confidence interval. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

A.Lower bound:

Upper bound:

(Round to three decimal places as needed.)

B. The requirements for constructing a confidence interval are not satisfied.

(2) Construct the 99% confidence interval. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

A.Lower bound:

Upper bound:

(Round to three decimal places as needed.)

B. The requirements for constructing a confidence interval are not satisfied.

(3) Choose the correct answer below.

A.Increasing the level of confidence has no effect on the interval.

B.Increasing the level of confidence widens the interval.

C.Increasing the level of confidence narrows the interval.

D.It is not possible to tell the effect of increasing the level of confidence on the width of the interval since the requirements for constructing a confidence interval in parts (b) and (c) were not met

A researcher wishes to estimate the percentage of adults who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 3 percentage points with 90% confidence if

(a) he uses a previous estimate of 36%?

(b) he does not use any prior estimates?

Explanation / Answer

Solution:

An interactive poll found that 331 of 2,297adults aged 18 or older have at least one tattoo.

Confidence Interval For Proportion
CI = p ± Z a/2 Sqrt(p*(1-p)/n)))
x = Mean
n = Sample Size
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
1)

Confidence Interval = [ 0.1441 ±Z a/2 ( Sqrt ( 0.1441*0.8559) /2297)]
=[0.1441±1.645 ( Sqrt ( 0.1441*0.8559) /2297)]   
=[0.1441±1.645 ( Sqrt ( 0.1441*0.8559) /2297)]
=[0.1441±1.645(0.00732)
upper bound=0.1561
lower bound=0.1320   

2)
Confidence Interval = [ 0.1441 ±Z a/2 ( Sqrt ( 0.1441*0.8559) /2297)]
=[ 0.1441 ±2.326 ( Sqrt ( 0.1441*0.8559) /2297)]
=[ 0.1441 ±2.326 ( Sqrt ( 0.1441*0.8559) /2297)]
=[ 0.1441 ±2.326 (0.00732)]
=[ 0.1441 ± 0.01702]   
upper bound=0.1611
lower bound=0.1270

3) Increasing the effect of level of confidence  narrows the interval( it is wider)

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