Explain the Chi – Square test.

Step 1
Introduction:
The chi-square test can be used to test the in the following scenarios:
To test the Goodness of fit of the variables when their expected and observed frequencies are given.
To test the independence of the categorical variables by making it into a contingency table.
To test the significance of the single variance with the given variance.
1). Goodness of fit:
Goodness of fit test is applied to check how well the sample data obtained fits the distribution of the selected population. It can also be viewed as whether the frequency distribution fits the given pattern. Most commonly used test to check the goodness of fit is the chi-square test.
There are two values involved are observed and the expected values. The observed value represents the frequency of particular category in the sample and the expected value is obtained from the given distribution. Moreover, it summarizes the difference between the expected and observed values of the given data.
The hypotheses are stated as given below:
Null hypothesis: Data comes from the specified distribution.
Alternative hypothesis: Data does not come from the specified distribution.
The chi-square test statistic is calculated using the formula given below:
${\displaystyle x^2=\sum \frac{{(O_i-E_i)}^2}{E_i}}$
where,
${\displaystyle O_i}$-Represents the observed values
${\displaystyle E_i}$-Represents the expected values
Step 2
2). Test for independence:
In test for independence, we test whether there is an association between the categorical variables.
Null hypothesis: There is no association between the two categorical variables.
Alternative hypothesis: There is no association between the two categorical variables.
The chi-square test statistic is calculated using the formula given below:
${\displaystyle x^2=\sum \sum \frac{{(O_{ij}-E_{ij})}^2}{E_{ij}}}$
where,
${\displaystyle O_{ij}}$-Represents the observed values
in $i^{th}\text{column and}{j}^{th}$ row
${\displaystyle E_{ij}}$-Represents the expected values
in $i^{th}\text{column and}{j}^{th}$ row
Step 3
3). Test for single variance:
Here, the chi-square test is used to compare the single sample variance with the population variance.
Null hypothesis: The given sample variance is equal to the population variance.
Alternative hypothesis: The given sample variance is not equal (less or greater) to the population variance.
The chi-square test statistic is calculated using the formula given below:
${\displaystyle x^2=\frac{(n-1)s^2}{{\sigma }^2}}$
where,
n-Sample size
${\displaystyle s^2}$-Sample variance
${\displaystyle {\sigma }^2}$-Population variance
Step 4
Examples for some research questions based on chi square tests:
1.Is there significant association between blood pressure and exercise level?
2.Whether the class has less variation in statistics marks than the other past statistics classes marks?
3.To check whether health insurance benefits vary by the size of the company.
4.To test the claim that the subject distribution of books in the library fits the distribution of books checked out by students.
5.The level of education and the amount of proceeded foods in an individual’s diet are independent or not.