A local service agency hires you to estimate the average annual income of all 700 families living in a four square block section. Since time and funds are limited, yu take a random sample of 50 families and want the estimate to have a 92% confidence level. The 50 families had an average income of $35,300. Research shows the population standard deviation for annual income in this four square block section is $1,800

Respuesta :

Answer:

The true estimate of  the average annual income of all 700 families living in a four square block section lies within the 92% confidence interval below

   [tex] 34854.3  <  \mu < 35 745.7[/tex]

Step-by-step explanation:

From the question we are told that

    The  sample size is  n =  50  

    The sample mean is [tex]\= x = \$ 35300[/tex]

     The population standard deviation is  [tex]\sigma = \$ 1800[/tex]

From the question we are told the confidence level is  92% , hence the level of significance is    

      [tex]\alpha = (100 - 92 ) \%[/tex]

=>   [tex]\alpha = 0.08[/tex]

Generally from the normal distribution table the critical value  of  [tex]\frac{\alpha }{2}[/tex] is  

   [tex]Z_{\frac{\alpha }{2} } =  1.751 [/tex]

Generally the margin of error is mathematically represented as  

      [tex]E = Z_{\frac{\alpha }{2} } *  \frac{\sigma }{\sqrt{n} }[/tex]

=>   [tex]E = 1.751  *  \frac{1800 }{\sqrt{50} }[/tex]

=>   [tex]E =  445.7 [/tex]

Generally 92% confidence interval is mathematically represented as  

      [tex]\= x -E <  \mu <  \=x  +E[/tex]

=> [tex] 35 300 -445.7 <  \mu < 35 300 + 445.7[/tex]

=> [tex] 34854.3  <  \mu < 35 745.7[/tex]

So the true estimate of  the average annual income of all 700 families living in a four square block section lies within the 92% confidence interval

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