Question

Let the random variable X follow a Normal distribution with variance σ² = 625.

A random sample of n = 50 is obtained with a sample mean, X-Bar of 180.

What is the probability that μ is between 198 and 211?

What is Z-Score1 for μ greater than 198?

Answer 1

Table in MS-Excel
sigma2=	625
sigma=	25
sqrt(2)=	1.414214
n	50
sqrt(n)=	7.071068
xbar	180
xbar-198	-18
xbar-211	-31
(xbar-198)/(sigma/sqrt(n))	-5.09117
(xbar-211)/(sigma/sqrt(n))	-8.76812
P(Z<-8.768124)	9.08E-19
P(Z<-5.091169)	1.78E-07
ROUND(NORMSDIST(-8.768124),1)	0
ROUND(NORMSDIST(-5.091169),1)	0

In general,a z-score indicates how many standard deviations an element is from the mean. It can be calculated from the following formula: z = (X - μ) / σ

where, z is the z-score, X is the value of the element, μ is the population mean, and σ is the standard deviation.