Kindergarten children have heights that are approximately no

Kindergarten children have heights that are approximately normally distributed about a mean of 39.3 inches and a standard deviation of 2.3 inches. If a random sample of 21 is taken, what is the probability that the sample of kindergarten children has a mean height of less than 38.4 inches? (Give your answer correct to four decimal places.)

Solution

Let X be the random variable that Kindergarten children heights.

X ~ Normal(mean = 39.3 inches, sd = 2.3 inches)

n = sample size = 21

What is the probability that the sample of kindergarten children has a mean height of less than 38.4 inches?

We have to find P(Xbar < 38.4).

And the sampling distribution of Xbar is also Normal with,

mean = 39.3

and s = sd/sqrt(n)

s = 2.3 / sqrt(21) = 0.5019

Now convert Xbar = 38.4 into standard normal distribution.

z = (Xbar - mean) / (sd/sqrt(n)) =  (Xbar - mean) / s

z = (38.4 - 39.3) / 0.5019 = -1.7932

Now we have to find P(Z < -1.7932).

This probability we can find by usin EXCEL.

syntax :

=NORMSDIST(z)

where z = -1.7932

P(Z < -1.7932) = 0.0365

The probability that the sample of kindergarten children has a mean height of less than 38.4 inches is 0.0365.