Answer:
The correct option is c
Step-by-step explanation:
From the question we are told that
The mean is [tex]\mu = 185.6 \ mm[/tex]
The standard deviation is [tex]\sigma = 12.7 \ mm[/tex]
The sample size is n = 10
Generally the standard error of the mean is mathematically represented as
[tex]\sigma_{x} = \frac{\sigma}{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{ 12.7}{\sqrt{ 10 } }[/tex]
=> [tex]\sigma_{x} = 4.016[/tex]
Generally the probability that a random sample of size 10 from this population will have a mean less than than 180 is mathematically represented as
[tex]P(X < 180 ) = P(\frac{ X - \mu }{\sigma_{x} } < \frac{180 - 185.6}{4.016} )[/tex]
[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P(X < 180 ) = P(Z < - 1.394 )[/tex]
From the z table the area under the normal curve to the left corresponding to -1.394 is
[tex]P(X < 180 ) = P(Z < - 1.394 ) = 0.081659[/tex]
=> [tex]P(X < 180 ) = 0.0823[/tex]
