A survey​ asked, “How many tattoos do you currently have on your​ body?” Of the 1230 males​ surveyed, 176 responded that they had at least o

Question

A survey​ asked, “How many tattoos do you currently have on your​ body?” Of the 1230 males​ surveyed, 176 responded that they had at least one tattoo. Of the 1079 females​ surveyed, 141 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.

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Allison 2 months 2021-10-09T00:50:21+00:00 1 Answer 0 views 0

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    2021-10-09T00:51:39+00:00

    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

    p_{1}  = \frac{x}{n_{1} } = \frac{176}{1230} =0.1430

    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

    p_{2}  = \frac{x}{n_{2} } = \frac{141}{1079} = 0.1306

    q₂ = 1-p₂ = 1-0.1306 =0.8694

    The 95​% confidence interval for p₁-p₂

    (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})

    where

            se(p_{1}-p_{2}) = \sqrt{\frac{p_{1}q_{1}  }{n_{1} }+\frac{p_{2} q_{2} }{n_{2} }  }

    se(p_{1}-p_{2}) = \sqrt{\frac{0.143(0.857)  }{1230}+\frac{ 0.1306(0.8694) }{1079 }

     se(p₁-p₂) = 0.01431

    (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})

    [(0.1430-0.1306) - 1.96(0.01431) , 0.1430-0.1306) + 1.96(0.01431)

    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 )

         

                                                     

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