Question

A population of values has a normal distribution with μ=240μ=240 and σ=77.2σ=77.2. You intend to draw a random sample of size n=214n=214.

Find the probability that a single randomly selected value is greater than 239.5.
P(X > 239.5) =

Find the probability that a sample of size n=214n=214 is randomly selected with a mean greater than 239.5.
P(M > 239.5) =

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

i)

Given

μ=240

σ=77.2

Now

= 1 - 0.4976

= 0.5024

Therefore, the probability that a single randomly selected value is greater than 239.5 = 0.5024

ii)

The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution with mean= and standard deviation = , as the sample size gets larger.

i.e.,  

i.e.,

i.e,

Now

= 1 - 0.4641

= 0.5359

Therefore, the probability that a sample of size n=214 is randomly selected with a mean greater than 239.5 = 0.5359