Question

In: Statistics and Probability

an interactive poll found that 368 of 2,375 adults aged 18 or older have at least...

an interactive poll found that 368 of 2,375 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?

Solutions

Expert Solution

Solution :

Given that,

n = 2375

x = 368

= x / n = 368 / 2375 =0.155

1 - = 1 - 0.155 = 0.845

a ) At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z/2 = Z0.05 = 1.645

Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 1.645 * (((0.155 * 0.845) / 2375)

= 0.012

A 90 % confidence interval for population proportion p is ,

- E < P < + E

0.155 - 0.012 < p < 0.155 + 0.012

0.143 < p < 0.167

The 90% confidence interval for the population proportion p is : ( 0.143 , 0.167)

b ) At 99% confidence level the z is ,

= 1 - 99+% = 1 - 0.99 = 0.01

/ 2 = 0.01/ 2 = 0.005

Z/2 = Z0.005 = 2.576

Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 2.576 * (((0.155 * 0.845) / 2375)

= 0.019

A 99 % confidence interval for population proportion p is ,

- E < P < + E

0.155 - 0.019 < p < 0.155 + 0.019

0.136 < p < 0.174

The 99% confidence interval for the population proportion p is : ( 0.136 , 0.174)

(d) The effect of increasing the level of confidence on the increasing width of the interval


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