Respuesta :
Answer:
The 95% confidence interval for p₁-p₂
( -0.01564 ,0.04044 )
Step-by-step explanation:
Explanation:-
Given data Of the 1230 males surveyed, 176 responded that they had at least one tattoo
Given the first sample size 'n₁' = 1230
Given x = 176
The first sample proportion
[tex]p_{1} = \frac{x}{n_{1} } = \frac{176}{1230} =0.1430[/tex]
q₁ = 1-p₁ =1-0.1430 = 0.857
Given data Of the 1079 females surveyed, 141 responded that they had at least one tattoo
Given the second sample size n₂ = 1079
and x = 141
The second sample proportion
[tex]p_{2} = \frac{x}{n_{2} } = \frac{141}{1079} = 0.1306[/tex]
q₂ = 1-p₂ = 1-0.1306 =0.8694
The 95% confidence interval for p₁-p₂
[tex](p_{1} - p_{2} - Z_{\frac{\alpha }{2} } se(p_{1} - p_{2}) ,p_{1} - p_{2} + Z_{\frac{\alpha }{2} } se(p_{1} - p_{2})[/tex]
where
[tex]se(p_{1}-p_{2}) = \sqrt{\frac{p_{1}q_{1} }{n_{1} }+\frac{p_{2} q_{2} }{n_{2} } }[/tex]
[tex]se(p_{1}-p_{2}) = \sqrt{\frac{0.143(0.857) }{1230}+\frac{ 0.1306(0.8694) }{1079 }[/tex]
se(p₁-p₂) = 0.01431
[tex](p_{1} - p_{2} - Z_{\frac{\alpha }{2} } se(p_{1} - p_{2}) ,p_{1} - p_{2} + Z_{\frac{\alpha }{2} } se(p_{1} - p_{2})[/tex]
[tex][(0.1430-0.1306) - 1.96(0.01431) , 0.1430-0.1306) + 1.96(0.01431)[/tex]
On calculation , we get
( 0.0124- 0.0280476 ,0.0124+ 0.0280476)
( -0.01564 ,0.04044 )
Conclusion:-
The 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.
( -0.01564 ,0.04044 )
