Answer:
The value is [tex]P(| \^ p - p| < 0.05 ) = 0.9822 [/tex]
Step-by-step explanation:
From the question we are told that
The population proportion is [tex]p = 0.52[/tex]
The sample size is n = 563
Generally the population mean of the sampling distribution is mathematically represented as
[tex]\mu_{x} = p = 0.52[/tex]
Generally the standard deviation of the sampling distribution is mathematically evaluated as
[tex]\sigma = \sqrt{\frac{ p(1- p)}{n} }[/tex]
=> [tex]\sigma = \sqrt{\frac{ 0.52 (1- 0.52 )}{563} }[/tex]
=> [tex]\sigma = 0.02106 [/tex]
Generally the probability that the proportion of persons with a college degree will differ from the population proportion by less than 5% is mathematically represented as
[tex]P(| \^ p - p| < 0.05 ) = P( - (0.05 - 0.52 ) < \^ p < (0.05 + 0.52 ))[/tex]
Here [tex]\^ p[/tex] is the sample proportion of persons with a college degree.
So
[tex]P( - (0.05 - 0.52 ) < \^ p < (0.05 + 0.52 )) = P(\frac{[[0.05 -0.52]]- 0.52}{0.02106} < \frac{[\^p - p] - p}{\sigma } < \frac{[[0.05 -0.52]] + 0.52}{0.02106} )[/tex]
Here
[tex]\frac{[\^p - p] - p}{\sigma } = Z (The\ standardized \ value \ of\ (\^ p - p))[/tex]
=> [tex]P( - (0.05 - 0.52 ) < \^ p < (0.05 + 0.52 )) = P[\frac{-0.47 - 0.52}{0.02106 } < Z < \frac{-0.47 + 0.52}{0.02106 }][/tex]
=> [tex]P( - (0.05 - 0.52 ) < \^ p < (0.05 + 0.52 )) = P[ -2.37 < Z < 2.37 ][/tex]
=> [tex]P( - (0.05 - 0.52 ) < \^ p < (0.05 + 0.52 )) = P(Z < 2.37 ) - P(Z < -2.37 )[/tex]
From the z-table the probability of (Z < 2.37 ) and (Z < -2.37 ) is
[tex]P(Z < 2.37 ) = 0.9911[/tex]
and
[tex]P(Z < - 2.37 ) = 0.0089[/tex]
So
=>[tex]P( - (0.05 - 0.52 ) < \^ p < (0.05 + 0.52 )) =0.9911-0.0089[/tex]
=>[tex]P( - (0.05 - 0.52 ) < \^ p < (0.05 + 0.52 )) = 0.9822 [/tex]
=> [tex]P(| \^ p - p| < 0.05 ) = 0.9822 [/tex]
