When the chi squared statistic is used in testing hypothesis? Include underlying assumptions and the test statistic for testing hypothesis on a single

Chi-square statistic for testing single variance:
To determine the population variance of a single sample, use the Chi-square statistic.
The necessary assumptions for Chi-square test:
Simple random sampling should be used to obtain the sample.
The sample's source population ought to exhibit normal distribution.
The data should be continuous.
The results of the chi-square test are as follows:
${\displaystyle x^2=\frac{(n-1)s^2}{{\sigma }^2}}$
${\displaystyle n=}$ Sample size
${\displaystyle s^2=}$ Sample variance
${\displaystyle {\sigma }^2=}$ Population variance
Decision rule based on P-value approach for both directional and non-directional tests:
The level of significance is $\alpha .$
If P-value ${\displaystyle \le \alpha }$, then reject the null hypothesis ${\displaystyle H_0}$.
If P-value ${\displaystyle >\alpha }$, then fail to reject the null hypothesis ${\displaystyle H_0}$.
P-value:
The P­-value will be obtained from the chi-square distribution table based on the value of test statistic and the degrees of freedom ${\displaystyle \left(n–1\right)}$ and the type of hypothesis test (Two tailed, right tailed or left tailed).
Chi-square statistic for testing distribution:
To determine whether the sample data are consistent with the hypothesized distribution or not, the Chi-square goodness of fit test is used.
The chi-square goodness-of-fit test is used to test whether a sample of data comes from a population with a specific distribution. The chi-square goodness-of-fit can also be applied to discrete distributions
The necessary assumptions for Chi-square test for goodness of fit are given below:
The sample should be collected using simple random sampling.
The variable of interest must be categorical.
The expected value of each cell should not be less than 5.
Evidently, the test is to determine whether a sample of data comes from a population with a specific distribution.
Chi-square goodness of fit is a right tailed test. Therefore, it is a directional test.
The chi-square test statistic is obtained as given below:
${\displaystyle x^2=\frac{\sum _{i=1}^n{(O_i-E_i)}^2}{E_i}}$
${\displaystyle O_i=}$ Observed frenquency
${\displaystyle E_i=}$ Expected frequency
Decision rule based on P-value approach:
The level of significance is $\alpha .$
If P-value ${\displaystyle \le \alpha }$, then reject the null hypothesis ${\displaystyle H_0}$.
If P-value ${\displaystyle >\alpha }$, then fail to reject the null hypothesis $H_0$.
The P­-value will be obtained from the chi-square distribution table based on the value of test statistic and the degrees of freedom ${\displaystyle \left(n–1\right)}$ for the right tailed test.