Question

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.

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

The 95% interval for
`p_1 - p_2` is
`-0.0171 ,0.0411`

Option A is correct

Explanation:

From the question we are told that

The sample size of male is
`n_1 = 1211`

The number of males that said they have at least one tattoo is
`r = 182`

The sample size of female is
`n_2 = 1041`

The number of females that said they have at least one tattoo is
`k = 144`

Generally the sample proportion of male is

`\r p_1 = (r)/( n_1)`

substituting values

`\r p_1 = ( 182)/(1211)`

`\r p_1 = 0.1503`

Generally the sample proportion of female is

`\r p_2 = (k)/( n_2)`

substituting values

`\r p_2 = ( 144)/(1041)`

`\r p_2 = 0.1383`

Given that the confidence level is 95% then the level of significance is mathematically represented as

`\alpha =100-95`

`\alpha =5\%`

`\alpha =0.05`

Next we obtain the critical value of
`(\alpha )/(2)` from the normal distribution table , the value is

`Z_(\alpha )/(2) = 1.96`

Generally the margin of error is mathematically represented as

`E = Z_{(\alpha )/(2) } * \sqrt{(\r p_1 (1- \r p_1))/(n_1) + (\r p_2 (1- \r p_2))/(n_2) }`

substituting values

`E = 1.96 * \sqrt{( 0.1503 (1- 0.1503))/(1211) + (0.1383 (1- 0.1383))/(1041) }`

`E = 0.0291`

The 95% confidence interval is mathematically represented as

`(\r p_1 - \r p_2 ) - E < p_1-p_2 < (\r p_1 - \r p_2 ) + E`

substituting values

`(0.1503- 0.1383 ) - 0.0291 < p_1-p_2 < (0.1503- 0.1383 ) + 0.0291`

`-0.0171 < p_1-p_2 < 0.0411`

So the interpretation is that there is 95% confidence that the difference of the proportion is in the interval .So conclude that there is insufficient evidence of a significant difference in the proportion of male and female that have at least one tattoo

This because the difference in proportion is less than
`\alpha`