Question

A sample of size 18 will be drawn from a population with mean 4 and standard deviation 3.

(a) Is it appropriate to use the normal distribution to find probabilities for x?

(b) If appropriate find the probability that x will be greater than 3.

(c) If appropriate find the 20th percentile of x.

Answer 1

a) No, as we are not given anything about the shape of the distribution of other metrics like median, or mode, assuming that is a normal distribution would not be correct and we cannot use the same to compute the probability of x. It could be a skewed distribution and thus not following the normal distribution properties.

b) If we assume it to be a normal distribution, we have here:

The probability here is computed as:
P( X > 3)

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.3694 is the required probability here.

c) From standard normal tables, we have:
P(Z < -0.8415) = 0.2

Therefore the 20^th percentile value of X here is computed as:

Therefore 1.4755 is the required 20^th percentile value here.