**Definition of Z-Test**

**Z-test** is a parametric statistical
test generally used to test the hypothesis
regarding the equality of the mean(s) of Normal Populations provided the
population variance to be known. In this test the distribution of the test
statistic under the null hypothesis is said to follow a Standard Normal
Distribution that is with mean of the distribution as **0** and the variance
as **1**.

**Structure of Z-Test**

Let (x_{1}, x_{2},
………….., x_{n}) be a random sample of size n from a univariate normal
population.

To
test H_{0}: μ = μ_{0 }vs H_{1}:
μ ≠ μ_{0}.

The population variance is known σ = σ_{0}
(say).

**This
follows a N (0,1) distribution under H _{0}.**

Now
the test criterion is set as if observed **|Z|> c**, then the hypothesis
is rejected at **100α%** level of significance. Here **c** is
the ritical value of the distribution at the chosen level of significance
say **α**, i.e.** **

**Z-test** is also used to test the equality
of two means of two independent univariate normal variables with the population
variance to be known and equal.

Also for testing the location of
bivariate normal population. The testing procedure is bit similar to the
process stated above.

**The Graph of Z-Test at 5% Level of
Significance**

**Graphs of Z-Test for different alternatives**

**Example of Z-Test**

A
cigarette manufacturer sent to a laboratory presumably identical samples of
tobacco. They made five determinations of the nicotine content in milligrams as
follows: (i) 21, 25, 27, 18, and 29. Test whether the mean is equal to 26
assuming the variance as 2. (Assume normality)

Here n = 5

Mean = 24

SD = 4.472

The value of the test statistic is Z
= -2.24.

Hence |Z| = 2.24.

Assuming is as a two tailed test and
the level of the test as 5% we take the critical value c as

Therefore c = τ_{0.025} =
1.96.

Then |Z|>c and the hypothesis is
rejected at 5% level of significance and it is concluded that the mean nicotine
content is not 26.

**Z-Test vs t-Test**

**T-test
**is used for testing the hypotheses
regarding the means of normal populations when the population variance is
unknown.

It
is also used to test the equality of population proportion.

**Z-test**
in comparison with Student’s t-test is more convenient. It has different values
tabulated for different levels of significance not for different sample size
whereas in t-test we will have different critical value for different sample
size.

**Student’s Z-test**

- Used when you know the standard
deviation of the population (σ)

**Student’s
t-Test**

- Used when you only know the
standard deviation of a sample(s)
- Used if small sample size
- Can also be used for comparing
two samples

**Properties of Z-test**

Here we discuss some important
properties of the Z-test:

- Z-test gets the power analysis
of the test if it is available.
- It determines the Confidence
Interval for the unknown parameter and also gets an estimated value for
it.
- It also takes into account the
hypothesized value of the parameter in
the test and determines the significance of the null or alternative
hypothesis.
- From Z-test we can also
calculate the standard error of the estimated value.
- Another important property or
advantage of Z-test is that the distribution of the test statistic is
always known and easy to deal with.

**Other Uses of Z-test**

**Z-test**can also be used to test hypotheses regarding populations other than normal because of the Central Limit Theorem.**Central Limit Theorem**(**CLT**) states that any test statistic can be approximately normally distributed for large samples. Using CLT we can approximate any test statistic as a normal variable and use the**Z-test**to perform the hypothesis test.- This test is also used for
testing the maximum likelihood estimators for the parameters in
statistical models with parametric approach.
- In nonparametric tests,
sometimes the test statistic in Mann -Whitney
**U**test can be approximated using normal distribution and**Z-test**is used thereby.

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