What is the statistical test for data validation with an example, Chi-square, ANOVA test, Z statics, T statics, F statics, Hypothesis Testing?

Before discussing the different statistical tests, we need to get a clear
understanding of what a null hypothesis is. A null hypothesis proposes that
has no significant difference exists in the set of a given observation.

Null: Two samples’ mean is equal. Alternate: Two samples mean are not
equal.
For rejecting the null hypothesis, a test is calculated. Then the test statistic
is compared with a critical value, and if found to be greater than the critical
value, the hypothesis will be rejected.

Critical Value:-

Critical values are the point beyond which we reject the null hypothesis.
Critical value tells us, what is the probability of N number of samples,
belonging to the same distribution. Higher, the critical value means
lower the probability of N number of samples belonging to the same
distribution.

Critical values can be used to do hypothesis testing in the following way.
1. Calculate the test statistic
2. Calculate critical values based on the significance level alpha
3. Compare test statistics with critical values

IMP-If the test statistic is lower than the critical value, accept the hypothesis
or else reject the hypothesis.

Chi-Square Test:-

A chi-square test is used if there is a relationship between two categorical
variables. The Chi-Square test is used to determine whether there is a significant
difference between the expected frequency and the observed frequency in
one or more categories. Chi-square is also called the non-parametric test
as it will not use any parameter.

ANOVA test:-
ANOVA also called an analysis of variance, is used to compare multiples
(three or more) samples with a single test.
Useful when there are more than three populations. Anova compares the
variance within and between the groups of the population. If the variation is
much larger than the within variation, the means of different samples will
not be equal. If the between and within variations are approximately the
same size, then there will be no significant difference between sample
means. Assumptions of ANOVA: 1-All populations involved follow a normal
distribution. 2-All populations have the same variance (or standard
deviation). 3-The samples are randomly selected and independent of one
another.
ANOVA uses the mean of the samples or the population to reject or
support the null hypothesis. Hence it is called parametric testing.

Z Statics:-
In a z-test, the samples are assumed to be normally distributed. A z score is
calculated with population parameters as “population mean” and
“population standard deviation” and it is used to validate a hypothesis that
the sample drawn belongs to the same population.
The statistics used for this hypothesis testing is called z-statistic, the score
for which is calculated as z = (x — μ) / (σ / √n), where x= sample mean μ =
the population mean σ / √n = population standard deviation If the test statistic is
lower than the critical value, accept the hypothesis or else reject the
hypothesis.

T Statics:-

A t-test used to compare the mean of the given samples. Like z-test, t-test
also assumed a normal distribution of the samples. A t-test is used when
the population parameters (mean and standard deviation) are unknown.

There are three versions of the t-test
1. Independent samples t-test which compare means for two groups
2. Paired sample t-test which compares mean from the same group at
different times
3. Sample t-test, which tests the mean of the single group against the
known mean. The statistic for hypothesis testing is called t-statistic,
the score for which is calculated as t = (x1 — x2) / (σ / √n1 + σ / √n2),
where
x1 = It is mean of sample A, x2 = mean of sample B,
n1 = size of sample 1 n2 = size of sample 2

F Statics:-
The F-test is designed to test if the two population variances are equal. It
compares the ratio of the two variances. Therefore, if the variances are
equal, then the ratio of the variances will be 1.
The F-distribution is the ratio of two independent chi-square variables
divided by their respective degrees of freedom.
F = s1^2 / s2^2 and where s1^2 > s2^2.
If the null hypothesis is true, then the F test-statistic given above can be
simplified. This ratio of sample variances will be tested statistic used. If the
null hypothesis is false, then we will reject the null hypothesis that the ratio
was equal to 1 and our assumption that they were equal.