Question

An interactive poll found that 349 of 2,336 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 95% confidence interval for the proportion of adults with at least one tattoo.

(c) Construct a 98% 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

a) 0.149

b) (0.135,0.163)

c) (0.134,0.164)

Explanation:

We are given the following in the question:

a) Sample size, n = 2336

Number of people that have at least one tattoo, x = 349

Point Estimate:

`\hat{p} = (x)/(n) = (349)/(2336) = 0.149`

b) 95% confidence interval:

`\hat{p}\pm z_(stat)\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}`

`z_(critical)\text{ at}~\alpha_(0.05) = 1.96`

Putting the values, we get:

`0.149\pm 1.96(\sqrt{(0.149(1-0.149))/(2336)}) = 0.149\pm 0.014\\\\=(0.135,0.163)`

c) 98% confidence interval

`z_(critical)\text{ at}~\alpha_(0.02) = 2.05`

Putting the values, we get:

`0.149\pm 2.05(\sqrt{(0.149(1-0.149))/(2336)}) = 0.149\pm 0.015\\\\=(0.134,0.164)`

d) With increase in confidence level, the width of the confidence interval increases.