A population of values has a normal distribution with μ=238.7 and σ=23.7. You intend to draw a random sample of size n=158.

Find the probability that a single randomly selected value is less than 236.1.

Find the probability that a sample of size n=158 is randomly selected with a mean less than 236.1.

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer:

Given:

The population mean $(\mu )=238.7$

The population standard deviation $(\sigma )=23.7$

The sample size $(n)=158$

(A) The probability that a single randomly selected value is less than 236.1:

$P(x<236.1)=P(\frac{x–\mu }{\sigma }<\frac{236.1–238.7}{23.8})=P(z<-0.109)=0.4566$

(B) The probability that a sample of size n=158 is randomly selected with a mean less than 236.1:

$P(M<236.1)=P(\frac{M-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{236.1–238.7}{\frac{23.7}{\sqrt{158}}})=P(z<-1.379)=0.0839$