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 $(μ)=238.7$
The population standard deviation $(σ)=23.7$
The sample size $(n)=158$
(A) The probability that a single randomly selected value is less than 236.1:
$P(x < 236.1) = P\left(\frac{x – \mu}{\sigma} < \frac{236.1 – 238.7}{23.8}\right) = P(z < -0.109) = 0.4566$
(B) The probability that a sample of size n=158 is randomly selected with a mean less than 236.1:
$P(M < 236.1) = P\left(\frac{M- \mu}{\frac{\sigma}{\sqrt{n}}} < \frac{236.1 – 238.7}{\frac{23.7}{\sqrt{158}}}\right) = P(z < -1.379) = 0.0839$
