Complete Question
The complete question is shown on the first uploaded image
Answer:
The 95% interval for [tex]p_1 - p_2[/tex] is [tex]-0.0171 ,0.0411[/tex]
Option A is correct
Step-by-step explanation:
From the question we are told that
The sample size of male is [tex]n_1 = 1211[/tex]
The number of males that said they have at least one tattoo is [tex]r = 182[/tex]
The sample size of female is [tex]n_2 = 1041[/tex]
The number of females that said they have at least one tattoo is [tex]k = 144[/tex]
Generally the sample proportion of male is
[tex]\r p_1 = \frac{r}{ n_1}[/tex]
substituting values
[tex]\r p_1 = \frac{ 182}{1211}[/tex]
[tex]\r p_1 = 0.1503[/tex]
Generally the sample proportion of female is
[tex]\r p_2 = \frac{k}{ n_2}[/tex]
substituting values
[tex]\r p_2 = \frac{ 144}{1041}[/tex]
[tex]\r p_2 = 0.1383[/tex]
Given that the confidence level is 95% then the level of significance is mathematically represented as
[tex]\alpha =100-95[/tex]
[tex]\alpha =5\%[/tex]
[tex]\alpha =0.05[/tex]
Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table , the value is
[tex]Z_\frac{\alpha }{2} = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \sqrt{\frac{\r p_1 (1- \r p_1)}{n_1} + \frac{\r p_2 (1- \r p_2)}{n_2} }[/tex]
substituting values
[tex]E = 1.96 * \sqrt{\frac{ 0.1503 (1- 0.1503)}{1211} + \frac{0.1383 (1- 0.1383)}{1041} }[/tex]
[tex]E = 0.0291[/tex]
The 95% confidence interval is mathematically represented as
[tex](\r p_1 - \r p_2 ) - E < p_1-p_2 < (\r p_1 - \r p_2 ) + E[/tex]
substituting values
[tex](0.1503- 0.1383 ) - 0.0291 < p_1-p_2 < (0.1503- 0.1383 ) + 0.0291[/tex]
[tex]-0.0171 < p_1-p_2 < 0.0411[/tex]
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 [tex]\alpha[/tex]
