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

Answer:

a) 0.149

b) (0.135,0.163)

c) (0.134,0.164)        

Step-by-step 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:

$\hat{p} = \dfrac{x}{n} = \dfrac{349}{2336} = 0.149$

b) 95% confidence interval:

$\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

$z_{critical}\text{ at}~\alpha_{0.05} = 1.96$

Putting the values, we get:

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

c) 98% confidence interval

$z_{critical}\text{ at}~\alpha_{0.02} = 2.05$

Putting the values, we get:

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

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