Question

A population of values has a normal distribution with u = 229.4 and a = 67.4. You intend to draw a random sample of size n = 16. Find the probability that a single randomly selected value is greater than 212.6. POX > 212.6) = Find the probability that a sample of size n = 16 is randomly selected with a mean greater than 212.6. PM > 212.6) = Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or 2-scores rounded to 3 decimal places are accepted. 2 Calculator Submit Question

Answer 1

Solution

Back-up Theory

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ², then, Z = (X - µ)/σ ~ N(0, 1), i.e., Standard Normal Distribution and hence

P(X ≤ or ≥ t) = P[{(X - µ)/σ} ≤ or ≥ {(t - µ)/σ}] = P[Z ≤ or ≥ {(t - µ)/σ}] .………………..............................................…(1)

Probability values for the Standard Normal Variable, Z, can be directly read off from Standard Normal Tables … (1a)

or can be found using Excel Function: Statistical, NORMSDIST(z) which gives P(Z ≤ z) …..................................(1b)

X bar ~ N(µ, σ²/n),……………………………………………………………........................................................…….(2),

where X bar is average of a sample of size n from population of X.

Now to work out the solution,

Let X represent the values. Then. Given X ~ N(229, 67.4) ................................................................................... (3)

And so, if Xbar is the sample mean based on 16 values,

vide (2), Xbar ~ X ~ N(229, 16.85) ....................................................................................................................... (4)

Part (a)

Probability a single randomly selected value is greater than 212.6

= P(X > 212.6)

= P[Z > {(212.6 - 229)/67.4}] [vide (1) and (3)]

= P(Z > - 0.2433)

= 0.5962 [vide (1b)] Answer 1

Part (b)

Probability that a sample of size 16 is randomly selected with mean greater than 212.6

= P(Xbar > 212.6)

= P[Z > {(212.6 - 229)/16.85}] [vide (1) and (4)]

= P(Z > - 0.9733)

= 0.8348 [vide (1b)] Answer 2

DONE