Inferential Statistics
2026-02-28 13:37 Diff
https://brightchamps.com/en-us/math/data/inferential-statistics

There are mainly two types that researchers use to draw conclusions from small samples. They are called hypothesis testing and regression analysis. Let us take a closer look at these two types of inferential statistics. 
 

Inferential Statistics Hypothesis Testing Regression Analysis Z test Linear Regression F test Nominal Regression Anova test Logistics Regression Wilcoxon Signed Rank test Ordinal Regression Mann-whitney U test  

Hypothesis Testing: Inferential statistics includes testing hypotheses and drawing conclusions about a population using sample data. It involves creating a null hypothesis and an alternative hypothesis before conducting a statistical test.

A hypothesis distribution can be left-tailed, right-tailed, or two-tailed. The conclusions are made by using the test statistic’s value, the critical value, and the confidence intervals. Some important hypothesis tests used in inferential statistics are:

  • Z Test:  Z test is used when data has a normal distribution and a sample size of at least 30. It is applied if the sample mean matches the population mean when the population variance is known. The right-tailed hypothesis can be tested using the following setup:

           Null hypothesis: H0: μ = μ0

           Alternative hypothesis: H1: μ > μ0

           Test statistic: Z Test = (x̄ – μ) / (σ / √n)

           Here, x̄ = Sample mean

           μ = Population mean

           σ = Standard deviation of the population

           n = Sample size

One thing we should keep in mind is that we can reject the null hypothesis if the z statistic is greater than the z critical value. 

  • T Test: A 't' test is used when the data has a student 't' distribution and the sample size is less than 30. When the population variance is unknown, the sample and population mean are compared. The inferential statistics hypothesis test is as follows: 

           Null hypothesis: H0: μ = μ0

           Alternative hypothesis: H1: μ > μ0

           Test statistic: t = x̄ – μs / √n

           Here, x̄ = Sample mean

           μ = Population mean

           s = Standard deviation of the population

           n = Sample size

If the 't' statistic is greater than the 't' critical value, then the null hypothesis can be rejected. 

  • F Test: An F-test helps compare two samples or populations to see if there are any prominent differences. A right-tailed F-test can be set up as follows:

           Null hypothesis: H0: σ21 = σ22

           Alternative hypothesis: H1: σ21 > σ22

           Test statistic: f = σ21 / σ22

           Here, σ21 = Variance of the first population

           σ22 = Variance of the second population

If the f test statistic is greater than the crucial value, we can reject the null hypothesis. 

For example:

A school wants to know if the average math score of Class A differs from the school average. They take a sample of 25 students and use a T-test. The T-test helps decide if Class A really performs differently or if the difference is due to random chance.

Regression Analysis: Regression analysis expresses the relationship between two variables. It determines how one variable responds to another. Simple linear, multiple linear, nominal, logistic, and ordinal regression are a few of the numerous regression models that can be applied.  


A commonly used regression form in inferential statistics is linear regression. Linear regression analyzes how the dependent variable reacts to a unit change in the independent variable. Some crucial formulas for regression analysis in inferential statistics are as follows: 

With α and β as regression coefficients, the straight-line equation is expressed as y = α + βx

β = ∑(xi − x̄) (yi − ȳ) / ∑(xi − x̄)2

β = rxy σy / σx

α = ȳ − βx̄ 

Here, x̄ = The mean of the independent variable

σx = Standard deviation of the first data set

ȳ = The mean of the dependent variable

σy = Standard deviation of the second data set

Here, you have to mention hypothesis testing and regression analysis.