Answer:
a
[tex]P(X < 12) = 0.3660[/tex]
b
[tex]P(X < 12) = 0.1521[/tex]
Step-by-step explanation:
Considering question a
From the question we are told that
The mean is [tex]\mu = 12.01 \ oz[/tex]
The standard deviation is [tex]\sigma = 0.35 \ oz[/tex]
The standard error of mean is mathematically represented as
[tex]\sigma_{x} = \frac{\sigma}{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{0.35}{\sqrt{144} }[/tex]
=> [tex]\sigma_{x} = 0.02917[/tex]
Generally the probability that the mean volume of a random sample of 144 bottles is less than 12 oz is mathematically represented as
[tex]P(X < 12) = P(\frac{X - \mu }{\sigma_{x}} < \frac{12 - 12.01 }{0.02917 } )[/tex]
[tex]\frac{X -\mu}{\sigma_{x} } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P(X < 12) = P(Z < -0.3425 )[/tex]
From the z table the area under the normal curve to the left corresponding to -0.3425 is
[tex]P(X < 12) = P(Z < -0.3425 ) = 0.3660[/tex]
=> [tex]P(X < 12) = 0.3660[/tex]
Considering question a
From the question we are told that
The mean is [tex]\mu = 12.03 \ oz[/tex]
Generally the probability that the mean volume of a random sample of 144 bottles is less than 12 oz is mathematically represented as
[tex]P(X < 12) = P(\frac{X - \mu }{\sigma_{x}} < \frac{12 - 12.03 }{0.02917 } )[/tex]
[tex]P(X < 12) = P(Z < -1.0274 )[/tex]
From the z table the area under the normal curve to the left corresponding to -1.0274 is
[tex]P(X < 12) = P(Z < -1.0274 ) =0.1521[/tex]
=> [tex]P(X < 12) = 0.1521[/tex]
