Question

1/ A physical therapist wants to determine the difference in the proportion of men and women who participate in regular sustained physical activity. What sample size should be obtained if she wishes the estimate to be within five percentage points with 95​% confidence, assuming that

​(a)she uses the estimates of 21.8​% male and 18.3​% female from a previous​ year? n = (Round up to the nearest whole​ number.)

​(b) she does not use any prior​ estimates?

2/ A survey​ asked, "How many tattoos do you currently have on your​ body?" Of the 1216 males​ surveyed,187 responded that they had at least one tattoo. Of the 1023 females​ surveyed, 129 responded that they had at least one tattoo. Construct a 90​% 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 p1 represent the proportion of males with tattoos and p2 represent the proportion of females with tattoos. Find the 90​% confidence interval for p1−p2.

The lower bound is

The upper bound is

​(Round to three decimal places as​ needed.)

Answer 1

Ans:

1)Margin of error=0.05

a)

1.96*sqrt((0.218*(1-0.218)/n)+(0.183*(1-0.183)/n))=0.05

1.96*sqrt(0.32/n)=0.05

sqrt(0.32/n)=0.05/1.96

n=0.32*(1.96/0.05)^2

n=492

b)when no prior estimate use 0.5

1.96*sqrt((0.5*(1-0.5)/n)+(0.5*(1-0.5)/n))=0.05

1.96*sqrt(0.5/n)=0.05

sqrt(0.5/n)=0.05/1.96

n=0.5*(1.96/0.05)^2

n=768

2)

sample proportion for males=187/1216=0.1538

sample proportion for females=129/1023=0.1261

90% confidence interval for difference in proportions

=(0.1538-0.1261)+/-1.645*SQRT((0.1538*(1-0.1538)/1216)+(0.1261*(1-0.1261)/1023))

=0.028+/-0.024

=(0.004, 0.052)

As,both limits are positive or it does not includes 0,so we can say that the proportion of males that have at least one tattoo differs significantly from the proportion of females that have at least one tattoo.