Question

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?

Answer 1

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 = Z_0.05 = 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 = Z_0.005 = 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