Question

11.1.21 A survey​ asked, "How many tattoos do you currently have on your​ body?" Of the males​ surveyed, responded that they had at least one tattoo. Of the females​ surveyed, responded that they had at least one tattoo. Construct a ​% confidence interval to judge whether the proportion of males that have at least one tattoo differs significantly from the proportion of females that have at least one tattoo. Interpret the interval. Let represent the proportion of males with tattoos and represent the proportion of females with tattoos. Find the ​% confidence interval for . The lower bound is nothing. The upper bound is nothing. ​(Round to three decimal places as​ needed.)

Answer 1

Complete question :

The Harris Poll conducted a survey in which they asked, "How many tattoos do you currently have on your body?" Of the 1205 males surveyed, 181 responded that they had at least one tattoo. Of the 1097 females surveyed, 143 responded that they had at least one tattoo. Construct a 95% confidence interval to judge whether the proportion of males that have at least one tattoo differs significantly from the proportion of females that have at least one tattoo. Interpret the interval.

Answer:

(−0.0085 ; 0.0481)

Step-by-step explanation:

Given that :

n1 = 1205 ; x1 = 181 ; n2 = 1097 ; x2 = 143 ; α = 95%

Zα/2 = 1.96 ( Z table)

Confidence interval : (p1 - p2) ± E

E =Zα/2 * √[(p1q1/n1) + (p2q2/n2)]

p1 = x1 /n1 =181/1205 = 0.1502

q1 = 1 - p1 = 1 -0.1502 = 0.8498

p2 = x2/n2 = 143/1097 = 0.1304

q2 = 1 - p2 = 1 -0.1304 = 0.8696

E = 1.96 * √(0.0001059 + 0.0001033)

E = 0.0283

p1 - p2 = 0.1502 - 0.1304 = 0.0198

Lower boundary = 0.0198 - 0.0283 = −0.0085

Upper boundary = 0.0198 + 0.0283 = 0.0481

(−0.0085 ; 0.0481)