Question

difference between the Z used for normal distribution and Z used in hypothesis testing

Clear up a question that came up in class When we first dealt with the Z-valuethe formula was. But recently it has been. s/Vn What is the deal with the shift- explain why (as if to a high school student who is in a stats class) and support your explanation with example

Answer 1

$Z = \frac{X-\mu}{\sigma}$

we use this above formulae to standardize the random variable X to standard normal distribution N(0,1). Here X is a random variable which follows a normal distribution. We calculate Z- score to easily find out the probabilities using standard normal table.

Example:- For a normally distributed random variable X, with a population mean 234 and population standard deviation 17, what is the probability that x is lesser than 250.

In the above problem we have to convert the X into standard normal variate Z.

Z = (251 -234)/17 = 1

P(X<250) = P(Z<1) = 0.841.

In Hypothesis testing we calculate Z as a test statistic to find whether the sample mean is a particular value or not.

Example: For a normally distributed random variable X, with a population mean 234 and population standard deviation 17, sample size of 10 is taken, and the sample mean is 240. Test whether the sample mean represents the population mean or not.

test statistic $Z = \frac{X-\mu}{\sigma/\sqrt{n}}$

Z = 1.116

at 5% significance Z_0.025 = 1.96

Z<Z_0.025 hence we conclude the mean of sample represents the population mean.