Question

A population of values has a normal distribution with μ=236.3μ=236.3 and σ=20.7σ=20.7. You intend to draw a random sample of size n=17n=17.

Find the probability that a single randomly selected value is less than 224.3.
P(X < 224.3) =

Find the probability that a sample of size n=17n=17 is randomly selected with a mean less than 224.3.
P(M < 224.3) =

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

SOLUTION:

From given data,

A population of values has a normal distribution with μ=236.3 and σ=20.7 You intend to draw a random sample of size n=17.

Where,

Mean = μ = 236.3

standard deviation = σ = 20.7

sample of size = n = 17

Z = (X - μ) / σ = (X - 236.3) / 20.7

Find the probability that a single randomly selected value is less than 224.3.

P(X < 224.3) = P ((X - μ) / σ < (224.3 - 236.3) / 20.7)

P(X < 224.3) = P ( Z < -0.57 )

P(X < 224.3) = 0.2843 (Using z score table)

Find the probability that a sample of size n=17 is randomly selected with a mean less than 224.3.

= μ = 236.3

= / sqrt(n) = 20.7 / sqrt(17) = 5.020487

Z = (M - ) / = (M - 236.3) / 5.020487​​​​​​​

P(M < 224.3) = P ((M-) / <   (224.3-236.3) /5.020487​​​​​​​ )

P(M < 224.3) = P ( Z < -2.39​​​​​​​ )

P(M < 224.3) = 0.0084  (Using z score table)