Complete Question
The complete question is shown on the first uploaded image
Answer:
a
[tex]P(0.51 < \^ p < 0.61 ) = 0.2587 [/tex]
b
[tex]P(\^ p > 0.71 ) = 0.15165[/tex]
Step-by-step explanation:
From the question we are told that
The population proportion is [tex]p = 0.64[/tex]
The sample size is n = 50
Generally the mean of this sampling distribution is
[tex]\mu_{x} = p = 0.64[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{ \frac{p(1- p )}{n } } [/tex]
=> [tex]\sigma = \sqrt{ \frac{0.64(1- 0.64 )}{ 50} } [/tex]
=> [tex]\sigma = 0.068 [/tex]
Generally the probability that the value of [tex]\^ p[/tex] will be between 0.54 and 0.61 is mathematically represented as
[tex]P(0.51 < \^ p < 0.61 ) = P( \frac{0.5 4 - \mu_{x}}{\sigma } < \frac{\^ p- \mu_{x}}{\sigma } < \frac{0.6 1 - \mu_{x}}{\sigma } )[/tex]
[tex]\frac{\^ p -\mu}{\sigma } = Z (The \ standardized \ value\ of \ \^ p )[/tex]
[tex]P(0.51 < \^ p < 0.61 ) = P( \frac{0.5 4 - 0.64}{0.068 } < Z < \frac{0.6 1 - 0.64}{0.068} )[/tex]
[tex]P(0.51 < \^ p < 0.61 ) = P(-1.470 < Z < -0.4412 )[/tex]
=> [tex]P(0.51 < \^ p < 0.61 ) = P( Z < -0.4412 ) - P(Z < -1.470 ) [/tex]
From the z table the probabilities of ( Z < -0.4412 ) and (Z < -1.912 ) is
[tex]P ( Z < -0.4412 ) = 0.32953 [/tex]
and
[tex]P(Z < -1.470 ) = 0.070781[/tex]
Generally
[tex]P(0.51 < \^ p < 0.61 ) = 0.32953 -0.070781 [/tex]
[tex]P(0.51 < \^ p < 0.61 ) = 0.2587 [/tex]
Generally the probability that the value of [tex]\^ p[/tex] will be greater than 0.71 is mathematically represented as
[tex]P(\^ p > 0.71 ) = P( \frac{\^ p - \mu_{x}}{\sigma} > \frac{0.71 - 0.64 }{ 0.068 } )[/tex]
=> [tex]P(\^ p > 0.71 ) = P( Z > 1.0294 )[/tex]
From the z table the probabilities of ( Z > 1.0294 )
[tex]P( Z > 1.0294 ) = 0.15165[/tex]
So
[tex]P(\^ p > 0.71 ) = 0.15165[/tex]
