Respuesta :
Answer:
a
[tex]P(X > 75)= 0.35402[/tex]
b
[tex]P(72 < X < 75 ) = 0.2529[/tex]
c
[tex]P( X < 74.7) = 0.74041[/tex]
Step-by-step explanation:
From the question we are told that
The population mean is [tex]\mu = 74[/tex]
The population standard deviation is [tex]\sigma = 16[/tex]
Considering question a
The sample size is n = 36
Generally the standard error of mean is mathematically represented as
[tex]\sigma_{x} = \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{16}{\sqrt{36} }[/tex]
=> [tex]\sigma_{x} = 2.67[/tex]
Generally the probability that a random sample of size 36 yielding a sample mean of 75 or more is mathematically represented as
[tex]P(X > 75) = P( \frac{X - \mu }{ \sigma_{x}} > \frac{75 - 74}{ 2.67 } )[/tex]
[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P(X > 75) = P(Z > 0.3745 )[/tex]
From the z table the area under the normal curve representing 0.3745 to the right is
[tex]P(Z > 0.3745 ) = 0.35402[/tex]
=> [tex]P(X > 75)= 0.35402[/tex]
Considering question b
The sample size is n = 104
Generally the standard error of mean is mathematically represented as
[tex]\sigma_{x} = \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{16}{\sqrt{104} }[/tex]
=> [tex]\sigma_{x} = 1.5689[/tex]
Generally the probability that a random sample of size 104 yielding a sample mean between 72 and 75 is mathematically represented as
[tex]P(72 < X < 75 ) = P(\frac{72 - 74 }{1.5689} < \frac{X - \mu }{\sigma_{x}} < \frac{75 - 74 }{1.5689} )[/tex]
=> [tex]P(72 < X < 75 ) = P(-1.275 < Z < 0.375 )[/tex]
=> [tex]P(72 < X < 75 ) = P(Z < 0.375 ) - P(Z < -1.275)[/tex]
From the z table the area under the normal curve representing -1.275 to to the left is
[tex]P(Z < -1.275) =0.10115[/tex]
=> [tex]P(72 < X < 75 ) = 0.35402 - 0.10115[/tex]
=> [tex]P(72 < X < 75 ) = 0.2529[/tex]
Considering question c
The sample size is n = 217
Generally the standard error of mean is mathematically represented as
[tex]\sigma_{x} = \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{16}{\sqrt{217} }[/tex]
=> [tex]\sigma_{x} = 1.086[/tex]
Generally the probability that a random sample of size 217 yielding a sample mean of less than 74.7 is mathematically represented as
[tex]P( X < 74.7) = P(\frac{X - \mu }{\sigma_x} < \frac{ 74.7 - 74 }{ 1.086 })[/tex]
=> [tex]P( X < 74.7) = P(Z < 0.6446 )[/tex]
From the z table the area under the normal curve representing 0.6446 to to the left is
[tex]P(Z < 0.6446 ) = 0.74041[/tex]
=> [tex]P( X < 74.7) = 0.74041[/tex]
