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According to a study done by the Pew Research Center, 39% of adult Americans believe that marriage is now obsolete. (a) Suppose a random sample of 500 adult Americans is asked whether marriage is obsolete. Describe the sampling distribution of p ˆ , the proportion of adult Americans who believe marriage is obsolete, by answering the following questions: (i) Verify the model requirements. i.e. verify that np(1 – p

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Answer:

The proportion follows normal distribution,

 Mean u = 0.39

 standard deviation σ = 0.0218

Step-by-step explanation:

Solution:-

- Lets assume the population proportion ( p ) to be the percentage of adult Americans who believe that marriage is now obsolete.

- We will check for normality:

 Lets take,   p^ = p  (Mean proportion of the distribution)

- The condition of normality,

                  n*p*( 1 - p ) ≥ 10

Where, n : The sample size taken.

                  500*0.39*( 1 - 0.39 ) = 500*0.39*( 0.61 )

                  118.95 ≥ 10

- Hence, with the testing statistics the condition for normality are validated. Hence, the distribution for proportion of adult americans who believe that marriage is now obsolete follows a normal distribution.

- The parameters of the distribution are:

    Mean : u = p^ = p = 0.39

    Standard deviation σ =  [tex]\sqrt{\frac{p*(1-p)}{n} }[/tex] =    [tex]\sqrt{\frac{0.39*(1-0.39)}{500} } = \sqrt{\frac{0.39*(0.61)}{500} } =0.0218[/tex]  

             

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