An interactive poll found that 386 of 2340 adults aged 18 or older have at least

Question

An interactive poll found that 386 of 2340 adults aged 18 or older have at least one tattoo. (a) Obtain a point estimate for the proportion of adults who have at least one tattoo. (b) Construct a 90% confidence interval for the proportion of adults with at least one tattoo. (c) Construct a 99% confidence interval for the proportion of adults with at least one tattoo.

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

A.

Increasing the level of confidence narrows the interval.

B.

Increasing the level of confidence widens the interval.

C.

Increasing the level of confidence has no effect on 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.

Explanation / Answer

a)

Note that              
              
p^ = point estimate of the population proportion = x / n = 386/2340 =   0.164957265   [ANSWER]

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b)      
              
Note that              
              
p^ = point estimate of the population proportion = x / n =    0.164957265          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.007672416          
              
Now, for the critical z,              
alpha/2 =   0.05          
Thus, z(alpha/2) =    1.644853627          
Thus,              
Margin of error = z(alpha/2)*sp =    0.012620001          
lower bound = p^ - z(alpha/2) * sp =   0.152337264          
upper bound = p^ + z(alpha/2) * sp =    0.177577266          
              
Thus, the confidence interval is              
              
(   0.152337264   ,   0.177577266   ) [ANSWER]

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c)

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.164957265          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.007672416          
              
Now, for the critical z,              
alpha/2 =   0.005          
Thus, z(alpha/2) =    2.575829304          
Thus,              
Margin of error = z(alpha/2)*sp =    0.019762834          
lower bound = p^ - z(alpha/2) * sp =   0.145194431          
upper bound = p^ + z(alpha/2) * sp =    0.184720099          
              
Thus, the confidence interval is              
              
(   0.145194431   ,   0.184720099   ) [ANSWER]

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d)

As we can see, the margin of error increased when we increase the confidence level.

Hence,

OPTION B: Increasing the level of confidence widens the interval. [ANSWER]

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