A population of values has a normal distribution with μ = 135.4 and σ = 80.7. You intend to draw a random sample of size n = 12 .

Question

-Find the probability that a single randomly selected value is less than 207.6. 

P(X < 207.6) =

-Find the probability that a sample of size n=12n=12 is randomly selected with a mean less than 207.6. 

P(M < 207.6) =

-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 1

Hi Amina,

We can compute the Z statistic. First, for a sample of n = 1:

Z = (Xmean - µ) / (σ/ √n) = (207.6-135.4)/(80.7/1) = 0.895 ~ 0.9

P(X< 207.6) = P(Z <0.90) = 0.81594, which we got from a Z score table.

Next, for a sample of n = 12:

Z = (Xmean - µ) / (σ/ √n) = (207.6-135.4)/(80.7/√12) = 3.09 ~3.1

P(M <207.6) = P (Z < 3.1) = 0.99903

The answer makes sense. For a sample of n = 1, we have a high probability of its value being smaller than 207.6, which is almost as high as µ + σ. For a larger sample of n = 12, we have a probability close to 1 that the mean of the sample will be smaller than this relatively large value of 207.6.