What is Inferential Statistics?

Inferential statistics is a branch of statistics that involves using data from a sample to make inferences about a larger population. It is concerned with making predictions, generalizations, and conclusions about a population based on the analysis of a sample of data.

So, statistical inference is the branch of statistics concerned with drawing conclusions and or making decisions concerning a population based only on sample data.

Apart from inferential statistics, descriptive statistics forms another branch of statistics. Inferential statistics help to draw conclusions about the population while descriptive statistics summarizes the features of the data set.

• Inferential statistics encompasses two primary categories – hypothesis testing and regression analysis.
• It is crucial for samples used in inferential statistics to be an accurate representation of the entire population.
• This article will delve deeper into inferential statistics, exploring its various types, offering examples, and highlighting significant formulas.

Example of Inferential Statistics

Suppose you are cooking some recipe and you want to test it before serving to the guest to get an idea about the dish as a whole. You will never eat the full dish to get that idea. Rather you will taste very little portion of your dish with a spoon.

• So here you are only doing exploratory analysis to get idea what you cook with a sample in your hand.
• Next if you generalize that your dish required some extra sugar or salt then that making an inference.
• To get a valid and right inference your portion of dish that you tested should be representative of your sample. Otherwise conclusion will be wrong.

Main Goal of inferential statistics

The main goal of inferential statistics is to use the information gained from a sample to make inferences or predictions about a larger population with a certain level of confidence or probability. This can involve testing hypotheses about the relationship between variables or making predictions about future outcomes.

Inferential statistics uses a variety of techniques, such as hypothesis testing, confidence intervals, and regression analysis, to make inferences about a population. These techniques involve analyzing the data from a sample and using statistical tests to determine the likelihood that the observed results are due to chance or some other factor.

The results of inferential statistics are always presented with some degree of uncertainty or margin of error, as it is not possible to know the true characteristics of a population with absolute certainty based on a sample of data. However, inferential statistics provides a framework for making informed decisions based on the best available evidence.

Different Types of Inferential Statistics

Inferential statistics can be broadly categorized into two types:

• Hypothesis Testing
• Regression Analysis

Hypothesis Testing

Hypothesis testing involves using a sample of data to test a hypothesis about a population parameter. A hypothesis is a statement about a population parameter, such as the population mean or proportion, that the researcher wants to test. Hypothesis testing involves comparing the observed sample statistic to what would be expected if the null hypothesis (the statement being tested) were true. This comparison is used to determine whether there is enough evidence to reject the null hypothesis in favor of an alternative hypothesis.

Example of Hypothesis Testing

Suppose a car manufacturer claims that their cars have an average fuel efficiency of 30 miles per gallon (mpg). A researcher wants to test whether this claim is true.

• The null hypothesis (H0) is that the average fuel efficiency of the cars is 30 mpg 
• And the alternative hypothesis (Ha) is that the average fuel efficiency is different from 30 mpg.

The researcher takes a sample of 50 cars and measures their fuel efficiency. The sample has a mean fuel efficiency of 28 mpg and a standard deviation of 4 mpg.

To test the hypothesis, the researcher uses a t-test to compare the sample mean to the hypothesized population mean. The t-test produces a t-value of -2.5 and a p-value of 0.015.

The p-value is the probability of obtaining a sample mean as extreme or more extreme than the one observed, assuming the null hypothesis is true. In this case, the p-value is less than the significance level of 0.05, so the researcher rejects the null hypothesis and concludes that the average fuel efficiency of the cars is significantly different from 30 mpg.

The researcher may then report the results and recommend that the car manufacturer re-evaluate their claim or investigate potential factors that may be causing the lower-than-expected fuel efficiency.

Regression Analysis

Regression analysis is used to model the relationship between one or more independent variables and a dependent variable. The goal of regression analysis is to create a mathematical equation that can be used to predict the value of the dependent variable based on the values of the independent variables. Regression analysis is commonly used in fields such as economics, psychology, and social sciences to understand how changes in one variable can affect another variable.

Example of Regression Analysis

Suppose you’re a real estate agent and you want to predict the selling price of a house based on its size (in square feet) and the number of bedrooms it has. To do this, you collect data on recently sold houses in a particular neighborhood, including the selling price, size, and number of bedrooms for each house. You can then use regression analysis to build a model that predicts the selling price of a house based on its size and number of bedrooms.

You might start by using simple linear regression to build a model with just one predictor variable (size or number of bedrooms). You would plot the data and fit a line to the points that represents the relationship between the predictor variable and the outcome variable (selling price). You could then use this line to predict the selling price of a new house based on its size or number of bedrooms. This an simplest example of Regression Analysis.

However, you might find that a more accurate model can be built by using multiple regression, which allows you to include both predictor variables (size and number of bedrooms) in the same model. Multiple regression involves fitting a plane or hyperplane to the data that takes into account the relationships between all the predictor variables and the outcome variable. This would allow you to predict the selling price of a new house based on both its size and number of bedrooms.

Other Types of Inferential Statistics

Other types of inferential statistics include analysis of variance (ANOVA), correlation analysis, and factor analysis.

ANOVA

ANOVA is used to test whether there are significant differences between the means of three or more groups.

Correlation Analysis

Correlation analysis is used to measure the strength and direction of the relationship between two variables.

Factor Analysis

Factor analysis is used to identify underlying factors or dimensions that explain the correlations between multiple variables.

 

What is Z Test?

A Z-test is a statistical test that is used to determine whether two population means are significantly different from each other when the variances of the populations are known. It is a parametric test that assumes a normal distribution of the data and is used when the sample size is large (typically more than 30).

The Z-test is used to test a hypothesis about the mean of a population based on a sample.

• Null hypothesis (H0): μ=μ0 The null hypothesis states that the mean of the population is equal to a specific value
• Alternative hypothesis (H1): μ>μ0 The alternative hypothesis states that the mean of the population is not equal to that specific value.

Z = (x – μ) / (σ / sqrt(n))

where:

• Z is the test statistic
• x is the sample mean
• μ is the hypothesized population mean
• σ is the population standard deviation
• n is the sample size

From the above formula, we can see that to perform a Z-test, the test statistic is calculated by subtracting the hypothesized population mean from the sample mean and dividing the result by the standard deviation of the sample. The resulting value is then compared to a critical value obtained from a Z-table or a p-value obtained from statistical software.

If the test statistic is greater than the critical value or if the p-value is less than the significance level (typically 0.05), then the null hypothesis is rejected and it is concluded that there is a significant difference between the population mean and the hypothesized value. On the other hand, if the test statistic is less than the critical value or if the p-value is greater than the significance level, then the null hypothesis is not rejected and it is concluded that there is not enough evidence to conclude that there is a significant difference between the population mean and the hypothesized value.

What is T-Test?

A t-test is a statistical test used to determine if there is a significant difference between the means of two groups of data. It is a common test used in hypothesis testing, where a researcher wants to determine whether there is a difference between two sets of data.

There are two main types of t-tests:

• the independent samples t-test
• the paired samples t-test

The independent samples t-test

The independent samples t-test is used when the samples being compared are independent of each other (i.e., the data comes from two separate groups of individuals). The paired samples t-test is used when the samples being compared are related (i.e., the data comes from the same group of individuals at different points in time).

Now we know that, in a t-test, the hypothesis being tested is whether there is a significant difference between the means of two groups.

• The null hypothesis (H0): The null hypothesis (H0) states that there is no significant difference between the means of the two groups
• Alternative hypothesis (Ha): Alternative hypothesis (Ha) states that there is a significant difference between the means of the two groups.

The null hypothesis is typically represented as: H0: μ1 = μ2

where μ1 represents the mean of the first group and μ2 represents the mean of the second group.

The alternative hypothesis can take one of three forms, depending on the nature of the research question:

1. Ha: μ1 ≠ μ2 (two-tailed test, where the alternative hypothesis suggests that the means are different in either direction)
2. Ha: μ1 > μ2 (one-tailed test, where the alternative hypothesis suggests that the mean of the first group is greater than the mean of the second group)
3. Ha: μ1 < μ2 (one-tailed test, where the alternative hypothesis suggests that the mean of the first group is less than the mean of the second group)

The hypothesis is tested using the t-test, and if the result is statistically significant, the null hypothesis is rejected in favor of the alternative hypothesis, indicating that there is a significant difference between the means of the two groups.

The paired samples t-test

The t-test calculates a t-value, which is then compared to a critical value from a t-distribution table to determine if the difference between the means is statistically significant. The t-value is influenced by the size of the difference between the means, the sample size, and the variance of the data. A significant result indicates that the means are unlikely to be equal by chance and that the difference is likely due to a real effect.

The paired samples t-test is used to test whether there is a significant difference between two related groups of data, such as the same group measured at two different points in time.

• The null hypothesis (H0): The null hypothesis (H0) for the paired samples t-test is that there is no significant difference between the means of the two related groups,
• The Alternative hypothesis (Ha): The Alternative hypothesis (Ha is that there is a significant difference between the means of the two related groups.

The null hypothesis is typically represented as: H0: μ1 – μ2 = 0

where μ1 represents the mean of the first related group and μ2 represents the mean of the second related group.

The alternative hypothesis can take one of two forms, depending on the nature of the research question:

• Ha: μ1 – μ2 ≠ 0 (two-tailed test, where the alternative hypothesis suggests that the means are different in either direction)
• Ha: μ1 – μ2 > 0 (one-tailed test, where the alternative hypothesis suggests that the mean of the first group is greater than the mean of the second group)
• Ha: μ1 – μ2 < 0 (one-tailed test, where the alternative hypothesis suggests that the mean of the first group is less than the mean of the second group)

These hypotheses are tested using the paired samples t-test, and if the result is statistically significant, the null hypothesis is rejected in favor of the alternative hypothesis, indicating that there is a significant difference between the means of the two related groups.

What is F-Test?

The F-test is a statistical test used to determine whether two population variances are equal. It is also used to test the overall significance of a regression model, by comparing the explained variance of the model to the unexplained variance.

In the case of comparing two population variances, the F-test calculates the ratio of the variances of two samples, and compares it to the F-distribution. If the ratio is significantly different from 1, it suggests that the two populations have significantly different variances.

The null and alternative hypotheses for the F-test depend on the specific context in which it is being used.

Testing for equality of variances:

• Null hypothesis: The population variances are equal (i.e., σ₁² = σ₂²).
• Alternative hypothesis: The population variances are not equal (i.e., σ₁² ≠ σ₂²).

Note that the F-test is always a two-tailed test, meaning that the alternative hypothesis can be either “not equal” or “greater than or less than” the null hypothesis.

In the case of testing the overall significance of a regression model, the F-test compares the variation in the dependent variable that is explained by the regression model to the variation that is not explained by the model. If the explained variation is significantly larger than the unexplained variation, it suggests that the model is a good fit for the data.

Testing for overall significance of a regression model:

• Null hypothesis: The regression model has no significant effect on the dependent variable (i.e., all the regression coefficients are zero).
• Alternative hypothesis: The regression model has a significant effect on the dependent variable (i.e., at least one of the regression coefficients is non-zero).

Note that the F-test is always a two-tailed test, meaning that the alternative hypothesis can be either “not equal” or “greater than or less than” the null hypothesis.

In both cases, the F-test assumes that the populations are normally distributed and that the samples are independent. It is a common test in many fields, including economics, engineering, and psychology.

A confidence interval

A confidence interval is a range of values that is likely to contain the true value of a population parameter with a certain degree of confidence or probability. In other words, a confidence interval is an estimate of the range of values within which a population parameter, such as the mean or proportion, is expected to lie based on a sample of data.

Confidence Interval Example

For example, if we want to estimate the mean height of a population, we can take a sample of individuals and calculate the sample mean height. However, this sample mean may not be exactly equal to the true population mean. A confidence interval provides a range of values within which the population mean is likely to lie with a certain level of confidence.

Typically, confidence intervals are calculated at the 95% or 99% level of confidence, which means that there is a 95% or 99% chance that the true population parameter falls within the calculated interval.

The width of the confidence interval depends on several factors, including the sample size, the level of confidence, and the variability of the data. A larger sample size, a higher level of confidence, and lower variability will result in a narrower confidence interval, while a smaller sample size, a lower level of confidence, and higher variability will result in a wider confidence interval.

Regression Analysis Basics

Regression analysis is a statistical method used to determine the relationship between one or more independent variables and a dependent variable. It is a commonly used statistical technique in many fields such as economics, psychology, social sciences, and engineering.

The purpose of regression analysis is to estimate the strength and direction of the relationship between the dependent variable and one or more independent variables, as well as to predict the value of the dependent variable based on the values of the independent variables.

There are several types of regression analysis, including:

Simple linear regression

This type of regression involves only one independent variable and a single dependent variable. The relationship between the variables is assumed to be linear, meaning that the change in the dependent variable is proportional to the change in the independent variable.

In a simple linear regression model, there is only one predictor variable, and the regression coefficient is typically denoted by the symbol “b”. The regression equation for a simple linear regression model can be written as:

y = b0 + b1*x

where y is the response variable, x is the predictor variable, b0 is the intercept term (the value of y when x=0), and b1 is the regression coefficient (the slope of the line).

Multiple linear regression:

This type of regression involves two or more independent variables and a single dependent variable. The relationship between the variables is again assumed to be linear.

In multiple linear regression models, there are multiple predictor variables, and the regression equation becomes:

y = b0 + b1x1 + b2x2 + … + bk*xk

where y is the response variable, x1, x2, …, xk are the predictor variables, b0 is the intercept term, and b1, b2, …, bk are the regression coefficients associated with each predictor variable.

Logistic regression

This type of regression is used when the dependent variable is categorical (e.g., yes or no), rather than continuous. Logistic regression models the relationship between the independent variables and the probability of the dependent variable being in a particular category.

Polynomial regression

This type of regression is used when the relationship between the independent variable and dependent variable is not linear, but can be modeled as a polynomial function.

In general, regression analysis involves fitting a mathematical model to the data using statistical techniques, such as ordinary least squares, and then using the model to make predictions or draw conclusions about the relationship between the variables. It is a powerful tool for analyzing and understanding complex relationships between variables and is widely used in many different fields of research.

Inferential Statistics vs Descriptive Statistics

Descriptive statistics and inferential statistics are two branches of statistics that are used to analyze and interpret data in different ways.

Descriptive Statistics	Inferential Statistics
Descriptive statistics are used to summarize and describe the main features of a dataset, such as the mean, median, mode, range, and standard deviation. These statistics are used to provide a basic understanding of the data, to identify patterns and trends, and to communicate the main findings to others.	In contrast, inferential statistics are used to make predictions or generalizations about a population based on a sample of data. These statistics involve the use of hypothesis testing and statistical models to draw conclusions about the relationships between variables or to test the significance of a particular result.
In other words, descriptive statistics simply describe what is happening in the data, while inferential statistics try to make inferences about what is happening in the population based on the data. For example, let’s say a researcher wants to know the average age of all students in a particular school. They collect data on a sample of 100 students and calculate the mean age of this sample. This is an example of descriptive statistics.	On the other hand, if the researcher wants to know whether the average age of students in this school is significantly different from the average age of students in another school, they would need to use inferential statistics to draw conclusions about the two populations. They might use a hypothesis test to determine whether the difference between the means of the two samples is statistically significant.

 

In summary, descriptive statistics are used to summarize and describe data, while inferential statistics are used to make predictions or draw conclusions about a population based on a sample of data.