Question

a survey asked, \how many tattoos do you currently have on your body?\ of the 1212 men surveyed, 179 responded that they had at least one tattoo. of the 1068 women surveyed, 143 responded that they had at least one tattoo. construct a 90% confidence interval to judge whether the proportion of men that have at least one tattoo differs significantly from the proportion of women that have at least one tattoo. interpret the interval. let ( p_1 ) represent the proportion of men with tattoos and ( p_2 ) represent the proportion of women with tattoos. find the 90% confidence interval for ( p_1 - p_2 ). the lower bound is. the upper bound is. (round to three decimal places as needed.)

Explanation:

Step1: Calculate sample proportions

For men: $\hat{p}_1=\frac{179}{1212}\approx0.148$
For women: $\hat{p}_2=\frac{143}{1068}\approx0.134$

Step2: Calculate the standard error

$SE = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1}+\frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$
$n_1 = 1212$, $n_2=1068$
$\hat{p}_1(1 - \hat{p}_1)=0.148\times(1 - 0.148)=0.148\times0.852 = 0.126$
$\hat{p}_2(1 - \hat{p}_2)=0.134\times(1 - 0.134)=0.134\times0.866=0.116$
$SE=\sqrt{\frac{0.126}{1212}+\frac{0.116}{1068}}=\sqrt{0.000104 + 0.000109}=\sqrt{0.000213}\approx0.015$

Step3: Find the critical value

For a 90% confidence interval, $\alpha=0.1$, $z_{\alpha/2}=z_{0.05} = 1.645$

Step4: Calculate the confidence interval

The confidence interval for $p_1 - p_2$ is $(\hat{p}_1-\hat{p}_2)\pm z_{\alpha/2}\times SE$
$\hat{p}_1-\hat{p}_2=0.148 - 0.134=0.014$
Lower bound: $0.014-1.645\times0.015=0.014 - 0.025=- 0.011$
Upper bound: $0.014 + 1.645\times0.015=0.014+0.025 = 0.039$

Answer:

The lower bound is $-0.011$
The upper bound is $0.039$