A sample of size 50 will be drawn from a population with a mean 76 and standard deviation 14. Find the 69th percentile of X

Question

A sample of size 50 will be drawn from a population with a mean 76 and standard deviation 14. Find the 69th percentile of X.

please be detail and step by step.

Answer 1

First step, find the Z score for the 65th percentile from a table (this is the inverse pdf )

Z crit = 0.3853

Second step, using the following equation, solve for x_i

Z = (x_i - μ )/ σ. for those algebraically challenged x = Zσ +μ. or .3853*14+76 = 81.394

NOTICE: since the question asked for the 65th percentile of X not the mean of X (Xbar) we don't need to calculate standard error. We use just the standard deviation. The sample size is not used.

Alternatively, use a computer or calculator to find the 65th percentile of a normally distributed variable with a mean of 76 and a sd of 14. For example, we can use the following r code.

Q.65 = qnorm(.65, mean = 76, sd = 14, lower.tail = TRUE)

The answer is 81.394 - surprise surprise