simple linear regression analysis hypothesis testing

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base
• Simple Linear Regression | An Easy Introduction & Examples

Simple Linear Regression | An Easy Introduction & Examples

Published on February 19, 2020 by Rebecca Bevans . Revised on June 22, 2023.

Simple linear regression is used to estimate the relationship between two quantitative variables . You can use simple linear regression when you want to know:

• How strong the relationship is between two variables (e.g., the relationship between rainfall and soil erosion).
• The value of the dependent variable at a certain value of the independent variable (e.g., the amount of soil erosion at a certain level of rainfall).

Regression models describe the relationship between variables by fitting a line to the observed data. Linear regression models use a straight line, while logistic and nonlinear regression models use a curved line. Regression allows you to estimate how a dependent variable changes as the independent variable(s) change.

If you have more than one independent variable, use multiple linear regression instead.

Table of contents

Assumptions of simple linear regression, how to perform a simple linear regression, interpreting the results, presenting the results, can you predict values outside the range of your data, other interesting articles, frequently asked questions about simple linear regression.

Simple linear regression is a parametric test , meaning that it makes certain assumptions about the data. These assumptions are:

• Homogeneity of variance (homoscedasticity) : the size of the error in our prediction doesn’t change significantly across the values of the independent variable.
• Independence of observations : the observations in the dataset were collected using statistically valid sampling methods , and there are no hidden relationships among observations.
• Normality : The data follows a normal distribution .

Linear regression makes one additional assumption:

• The relationship between the independent and dependent variable is linear : the line of best fit through the data points is a straight line (rather than a curve or some sort of grouping factor).

If your data do not meet the assumptions of homoscedasticity or normality, you may be able to use a nonparametric test instead, such as the Spearman rank test.

If your data violate the assumption of independence of observations (e.g., if observations are repeated over time), you may be able to perform a linear mixed-effects model that accounts for the additional structure in the data.

Here's why students love Scribbr's proofreading services

Discover proofreading & editing

Simple linear regression formula

The formula for a simple linear regression is:

• y is the predicted value of the dependent variable ( y ) for any given value of the independent variable ( x ).
• B 0 is the intercept , the predicted value of y when the x is 0.
• B 1 is the regression coefficient – how much we expect y to change as x increases.
• x is the independent variable ( the variable we expect is influencing y ).
• e is the error of the estimate, or how much variation there is in our estimate of the regression coefficient.

Linear regression finds the line of best fit line through your data by searching for the regression coefficient (B 1 ) that minimizes the total error (e) of the model.

While you can perform a linear regression by hand , this is a tedious process, so most people use statistical programs to help them quickly analyze the data.

Simple linear regression in R

R is a free, powerful, and widely-used statistical program. Download the dataset to try it yourself using our income and happiness example.

Dataset for simple linear regression (.csv)

Load the income.data dataset into your R environment, and then run the following command to generate a linear model describing the relationship between income and happiness:

This code takes the data you have collected data = income.data and calculates the effect that the independent variable income has on the dependent variable happiness using the equation for the linear model: lm() .

To learn more, follow our full step-by-step guide to linear regression in R .

To view the results of the model, you can use the summary() function in R:

This function takes the most important parameters from the linear model and puts them into a table, which looks like this:

This output table first repeats the formula that was used to generate the results (‘Call’), then summarizes the model residuals (‘Residuals’), which give an idea of how well the model fits the real data.

Next is the ‘Coefficients’ table. The first row gives the estimates of the y-intercept, and the second row gives the regression coefficient of the model.

Row 1 of the table is labeled (Intercept) . This is the y-intercept of the regression equation, with a value of 0.20. You can plug this into your regression equation if you want to predict happiness values across the range of income that you have observed:

The next row in the ‘Coefficients’ table is income. This is the row that describes the estimated effect of income on reported happiness:

The Estimate column is the estimated effect , also called the regression coefficient or r 2 value. The number in the table (0.713) tells us that for every one unit increase in income (where one unit of income = 10,000) there is a corresponding 0.71-unit increase in reported happiness (where happiness is a scale of 1 to 10).

The Std. Error column displays the standard error of the estimate. This number shows how much variation there is in our estimate of the relationship between income and happiness.

The t value  column displays the test statistic . Unless you specify otherwise, the test statistic used in linear regression is the t value from a two-sided t test . The larger the test statistic, the less likely it is that our results occurred by chance.

The Pr(>| t |)  column shows the p value . This number tells us how likely we are to see the estimated effect of income on happiness if the null hypothesis of no effect were true.

Because the p value is so low ( p < 0.001),  we can reject the null hypothesis and conclude that income has a statistically significant effect on happiness.

The last three lines of the model summary are statistics about the model as a whole. The most important thing to notice here is the p value of the model. Here it is significant ( p < 0.001), which means that this model is a good fit for the observed data.

When reporting your results, include the estimated effect (i.e. the regression coefficient), standard error of the estimate, and the p value. You should also interpret your numbers to make it clear to your readers what your regression coefficient means:

It can also be helpful to include a graph with your results. For a simple linear regression, you can simply plot the observations on the x and y axis and then include the regression line and regression function:

Prevent plagiarism. Run a free check.

No! We often say that regression models can be used to predict the value of the dependent variable at certain values of the independent variable. However, this is only true for the range of values where we have actually measured the response.

We can use our income and happiness regression analysis as an example. Between 15,000 and 75,000, we found an r 2 of 0.73 ± 0.0193. But what if we did a second survey of people making between 75,000 and 150,000?

The r 2 for the relationship between income and happiness is now 0.21, or a 0.21-unit increase in reported happiness for every 10,000 increase in income. While the relationship is still statistically significant (p<0.001), the slope is much smaller than before.

What if we hadn’t measured this group, and instead extrapolated the line from the 15–75k incomes to the 70–150k incomes?

You can see that if we simply extrapolated from the 15–75k income data, we would overestimate the happiness of people in the 75–150k income range.

If we instead fit a curve to the data, it seems to fit the actual pattern much better.

It looks as though happiness actually levels off at higher incomes, so we can’t use the same regression line we calculated from our lower-income data to predict happiness at higher levels of income.

Even when you see a strong pattern in your data, you can’t know for certain whether that pattern continues beyond the range of values you have actually measured. Therefore, it’s important to avoid extrapolating beyond what the data actually tell you.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).

A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.

Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative.

For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.

Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:

• measuring the distance of the observed y-values from the predicted y-values at each value of x;
• squaring each of these distances;
• calculating the mean of each of the squared distances.

Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Bevans, R. (2023, June 22). Simple Linear Regression | An Easy Introduction & Examples. Scribbr. Retrieved July 30, 2024, from https://www.scribbr.com/statistics/simple-linear-regression/

Is this article helpful?

Rebecca Bevans

Other students also liked, an introduction to t tests | definitions, formula and examples, multiple linear regression | a quick guide (examples), linear regression in r | a step-by-step guide & examples, what is your plagiarism score.

• Prompt Library
• DS/AI Trends
• Stats Tools
• Interview Questions
• Generative AI
• Machine Learning
• Deep Learning

Linear regression hypothesis testing: Concepts, Examples

In relation to machine learning , linear regression is defined as a predictive modeling technique that allows us to build a model which can help predict continuous response variables as a function of a linear combination of explanatory or predictor variables. While training linear regression models, we need to rely on hypothesis testing in relation to determining the relationship between the response and predictor variables. In the case of the linear regression model, two types of hypothesis testing are done. They are T-tests and F-tests . In other words, there are two types of statistics that are used to assess whether linear regression models exist representing response and predictor variables. They are t-statistics and f-statistics. As data scientists , it is of utmost importance to determine if linear regression is the correct choice of model for our particular problem and this can be done by performing hypothesis testing related to linear regression response and predictor variables. Many times, it is found that these concepts are not very clear with a lot many data scientists. In this blog post, we will discuss linear regression and hypothesis testing related to t-statistics and f-statistics . We will also provide an example to help illustrate how these concepts work.

Table of Contents

What are linear regression models?

A linear regression model can be defined as the function approximation that represents a continuous response variable as a function of one or more predictor variables. While building a linear regression model, the goal is to identify a linear equation that best predicts or models the relationship between the response or dependent variable and one or more predictor or independent variables.

There are two different kinds of linear regression models. They are as follows:

• Simple or Univariate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and one predictor or independent variable. The form of the equation that represents a simple linear regression model is Y=mX+b, where m is the coefficients of the predictor variable and b is bias. When considering the linear regression line, m represents the slope and b represents the intercept.
• Multiple or Multi-variate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and more than one predictor or independent variable. The form of the equation that represents a multiple linear regression model is Y=b0+b1X1+ b2X2 + … + bnXn, where bi represents the coefficients of the ith predictor variable. In this type of linear regression model, each predictor variable has its own coefficient that is used to calculate the predicted value of the response variable.

While training linear regression models, the requirement is to determine the coefficients which can result in the best-fitted linear regression line. The learning algorithm used to find the most appropriate coefficients is known as least squares regression . In the least-squares regression method, the coefficients are calculated using the least-squares error function. The main objective of this method is to minimize or reduce the sum of squared residuals between actual and predicted response values. The sum of squared residuals is also called the residual sum of squares (RSS). The outcome of executing the least-squares regression method is coefficients that minimize the linear regression cost function .

The residual e of the ith observation is represented as the following where [latex]Y_i[/latex] is the ith observation and [latex]\hat{Y_i}[/latex] is the prediction for ith observation or the value of response variable for ith observation.

[latex]e_i = Y_i – \hat{Y_i}[/latex]

The residual sum of squares can be represented as the following:

[latex]RSS = e_1^2 + e_2^2 + e_3^2 + … + e_n^2[/latex]

The least-squares method represents the algorithm that minimizes the above term, RSS.

Once the coefficients are determined, can it be claimed that these coefficients are the most appropriate ones for linear regression? The answer is no. After all, the coefficients are only the estimates and thus, there will be standard errors associated with each of the coefficients.  Recall that the standard error is used to calculate the confidence interval in which the mean value of the population parameter would exist. In other words, it represents the error of estimating a population parameter based on the sample data. The value of the standard error is calculated as the standard deviation of the sample divided by the square root of the sample size. The formula below represents the standard error of a mean.

[latex]SE(\mu) = \frac{\sigma}{\sqrt(N)}[/latex]

Thus, without analyzing aspects such as the standard error associated with the coefficients, it cannot be claimed that the linear regression coefficients are the most suitable ones without performing hypothesis testing. This is where hypothesis testing is needed . Before we get into why we need hypothesis testing with the linear regression model, let’s briefly learn about what is hypothesis testing?

Train a Multiple Linear Regression Model using R

Before getting into understanding the hypothesis testing concepts in relation to the linear regression model, let’s train a multi-variate or multiple linear regression model and print the summary output of the model which will be referred to, in the next section. 

The data used for creating a multi-linear regression model is BostonHousing which can be loaded in RStudioby installing mlbench package. The code is shown below:

install.packages(“mlbench”) library(mlbench) data(“BostonHousing”)

Once the data is loaded, the code shown below can be used to create the linear regression model.

attach(BostonHousing) BostonHousing.lm <- lm(log(medv) ~ crim + chas + rad + lstat) summary(BostonHousing.lm)

Executing the above command will result in the creation of a linear regression model with the response variable as medv and predictor variables as crim, chas, rad, and lstat. The following represents the details related to the response and predictor variables:

• log(medv) : Log of the median value of owner-occupied homes in USD 1000’s
• crim : Per capita crime rate by town
• chas : Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
• rad : Index of accessibility to radial highways
• lstat : Percentage of the lower status of the population

The following will be the output of the summary command that prints the details relating to the model including hypothesis testing details for coefficients (t-statistics) and the model as a whole (f-statistics) 

Hypothesis tests & Linear Regression Models

Hypothesis tests are the statistical procedure that is used to test a claim or assumption about the underlying distribution of a population based on the sample data. Here are key steps of doing hypothesis tests with linear regression models:

• Hypothesis formulation for T-tests: In the case of linear regression, the claim is made that there exists a relationship between response and predictor variables, and the claim is represented using the non-zero value of coefficients of predictor variables in the linear equation or regression model. This is formulated as an alternate hypothesis. Thus, the null hypothesis is set that there is no relationship between response and the predictor variables . Hence, the coefficients related to each of the predictor variables is equal to zero (0). So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis for each test states that a1 = 0, a2 = 0, a3 = 0 etc. For all the predictor variables, individual hypothesis testing is done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. Thus, if there are, say, 5 features, there will be five hypothesis tests and each will have an associated null and alternate hypothesis.
• Hypothesis formulation for F-test : In addition, there is a hypothesis test done around the claim that there is a linear regression model representing the response variable and all the predictor variables. The null hypothesis is that the linear regression model does not exist . This essentially means that the value of all the coefficients is equal to zero. So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis states that a1 = a2 = a3 = 0.
• F-statistics for testing hypothesis for linear regression model : F-test is used to test the null hypothesis that a linear regression model does not exist, representing the relationship between the response variable y and the predictor variables x1, x2, x3, x4 and x5. The null hypothesis can also be represented as x1 = x2 = x3 = x4 = x5 = 0. F-statistics is calculated as a function of sum of squares residuals for restricted regression (representing linear regression model with only intercept or bias and all the values of coefficients as zero) and sum of squares residuals for unrestricted regression (representing linear regression model). In the above diagram, note the value of f-statistics as 15.66 against the degrees of freedom as 5 and 194. 
• Evaluate t-statistics against the critical value/region : After calculating the value of t-statistics for each coefficient, it is now time to make a decision about whether to accept or reject the null hypothesis. In order for this decision to be made, one needs to set a significance level, which is also known as the alpha level. The significance level of 0.05 is usually set for rejecting the null hypothesis or otherwise. If the value of t-statistics fall in the critical region, the null hypothesis is rejected. Or, if the p-value comes out to be less than 0.05, the null hypothesis is rejected.
• Evaluate f-statistics against the critical value/region : The value of F-statistics and the p-value is evaluated for testing the null hypothesis that the linear regression model representing response and predictor variables does not exist. If the value of f-statistics is more than the critical value at the level of significance as 0.05, the null hypothesis is rejected. This means that the linear model exists with at least one valid coefficients. 
• Draw conclusions : The final step of hypothesis testing is to draw a conclusion by interpreting the results in terms of the original claim or hypothesis. If the null hypothesis of one or more predictor variables is rejected, it represents the fact that the relationship between the response and the predictor variable is not statistically significant based on the evidence or the sample data we used for training the model. Similarly, if the f-statistics value lies in the critical region and the value of the p-value is less than the alpha value usually set as 0.05, one can say that there exists a linear regression model.

Why hypothesis tests for linear regression models?

The reasons why we need to do hypothesis tests in case of a linear regression model are following:

• By creating the model, we are establishing a new truth (claims) about the relationship between response or dependent variable with one or more predictor or independent variables. In order to justify the truth, there are needed one or more tests. These tests can be termed as an act of testing the claim (or new truth) or in other words, hypothesis tests.
• One kind of test is required to test the relationship between response and each of the predictor variables (hence, T-tests)
• Another kind of test is required to test the linear regression model representation as a whole. This is called F-test.

While training linear regression models, hypothesis testing is done to determine whether the relationship between the response and each of the predictor variables is statistically significant or otherwise. The coefficients related to each of the predictor variables is determined. Then, individual hypothesis tests are done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. If at least one of the null hypotheses is rejected, it represents the fact that there exists no relationship between response and that particular predictor variable. T-statistics is used for performing the hypothesis testing because the standard deviation of the sampling distribution is unknown. The value of t-statistics is compared with the critical value from the t-distribution table in order to make a decision about whether to accept or reject the null hypothesis regarding the relationship between the response and predictor variables. If the value falls in the critical region, then the null hypothesis is rejected which means that there is no relationship between response and that predictor variable. In addition to T-tests, F-test is performed to test the null hypothesis that the linear regression model does not exist and that the value of all the coefficients is zero (0). Learn more about the linear regression and t-test in this blog – Linear regression t-test: formula, example .

Recent Posts

• Completion Model vs Chat Model: Python Examples - June 30, 2024
• LLM Hosting Strategy, Options & Cost: Examples - June 30, 2024
• Application Architecture for LLM Applications: Examples - June 25, 2024

Ajitesh Kumar

One response.

Very informative

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

• Search for:
• Excellence Awaits: IITs, NITs & IIITs Journey

ChatGPT Prompts (250+)

• Generate Design Ideas for App
• Expand Feature Set of App
• Create a User Journey Map for App
• Generate Visual Design Ideas for App
• Generate a List of Competitors for App
• Completion Model vs Chat Model: Python Examples
• LLM Hosting Strategy, Options & Cost: Examples
• Application Architecture for LLM Applications: Examples
• Python Pickle Security Issues / Risk
• Pricing Analytics in Banking: Strategies, Examples

Data Science / AI Trends

• • Prepend any arxiv.org link with talk2 to load the paper into a responsive chat application
• • Custom LLM and AI Agents (RAG) On Structured + Unstructured Data - AI Brain For Your Organization
• • Guides, papers, lecture, notebooks and resources for prompt engineering
• • Common tricks to make LLMs efficient and stable
• • Machine learning in finance

Free Online Tools

• Create Scatter Plots Online for your Excel Data
• Histogram / Frequency Distribution Creation Tool
• Online Pie Chart Maker Tool
• Z-test vs T-test Decision Tool
• Independent samples t-test calculator

Recent Comments

I found it very helpful. However the differences are not too understandable for me

Very Nice Explaination. Thankyiu very much,

in your case E respresent Member or Oraganization which include on e or more peers?

Such a informative post. Keep it up

Thank you....for your support. you given a good solution for me.

Linear regression - Hypothesis testing

by Marco Taboga , PhD

This lecture discusses how to perform tests of hypotheses about the coefficients of a linear regression model estimated by ordinary least squares (OLS).

Table of contents

Normal vs non-normal model

The linear regression model, matrix notation, tests of hypothesis in the normal linear regression model, test of a restriction on a single coefficient (t test), test of a set of linear restrictions (f test), tests based on maximum likelihood procedures (wald, lagrange multiplier, likelihood ratio), tests of hypothesis when the ols estimator is asymptotically normal, test of a restriction on a single coefficient (z test), test of a set of linear restrictions (chi-square test), learn more about regression analysis.

The lecture is divided in two parts:

in the first part, we discuss hypothesis testing in the normal linear regression model , in which the OLS estimator of the coefficients has a normal distribution conditional on the matrix of regressors;

in the second part, we show how to carry out hypothesis tests in linear regression analyses where the hypothesis of normality holds only in large samples (i.e., the OLS estimator can be proved to be asymptotically normal).

We also denote:

We now explain how to derive tests about the coefficients of the normal linear regression model.

It can be proved (see the lecture about the normal linear regression model ) that the assumption of conditional normality implies that:

How the acceptance region is determined depends not only on the desired size of the test , but also on whether the test is:

one-tailed (only one of the two things, i.e., either smaller or larger, is possible).

For more details on how to determine the acceptance region, see the glossary entry on critical values .

The F test is one-tailed .

A critical value in the right tail of the F distribution is chosen so as to achieve the desired size of the test.

Then, the null hypothesis is rejected if the F statistics is larger than the critical value.

In this section we explain how to perform hypothesis tests about the coefficients of a linear regression model when the OLS estimator is asymptotically normal.

As we have shown in the lecture on the properties of the OLS estimator , in several cases (i.e., under different sets of assumptions) it can be proved that:

These two properties are used to derive the asymptotic distribution of the test statistics used in hypothesis testing.

The test can be either one-tailed or two-tailed . The same comments made for the t-test apply here.

Like the F test, also the Chi-square test is usually one-tailed .

The desired size of the test is achieved by appropriately choosing a critical value in the right tail of the Chi-square distribution.

The null is rejected if the Chi-square statistics is larger than the critical value.

Want to learn more about regression analysis? Here are some suggestions:

R squared of a linear regression ;

Gauss-Markov theorem ;

Generalized Least Squares ;

Multicollinearity ;

Dummy variables ;

Selection of linear regression models

Partitioned regression ;

Ridge regression .

How to cite

Please cite as:

Taboga, Marco (2021). "Linear regression - Hypothesis testing", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/linear-regression-hypothesis-testing.

Most of the learning materials found on this website are now available in a traditional textbook format.

• F distribution
• Beta distribution
• Conditional probability
• Central Limit Theorem
• Binomial distribution
• Mean square convergence
• Delta method
• Almost sure convergence
• Mathematical tools
• Fundamentals of probability
• Probability distributions
• Asymptotic theory
• Fundamentals of statistics
• About Statlect
• Cookies, privacy and terms of use
• Loss function
• Almost sure
• Type I error
• Precision matrix
• Integrable variable
• To enhance your privacy,
• we removed the social buttons,
• but don't forget to share .

• Skip to secondary menu
• Skip to main content
• Skip to primary sidebar

Statistics By Jim

Making statistics intuitive

Linear Regression Explained with Examples

By Jim Frost 13 Comments

What is Linear Regression?

Linear regression models the relationships between at least one explanatory variable and an outcome variable. This flexible analysis allows you to separate the effects of complicated research questions, allowing you to isolate each variable’s role. Additionally, linear models can fit curvature and interaction effects.

Statisticians refer to the explanatory variables in linear regression as independent variables (IV) and the outcome as dependent variables (DV). When a linear model has one IV, the procedure is known as simple linear regression. When there are more than one IV, statisticians refer to it as multiple regression. These models assume that the average value of the dependent variable depends on a linear function of the independent variables.

Linear regression has two primary purposes—understanding the relationships between variables and prediction.

• The coefficients represent the estimated magnitude and direction (positive/negative) of the relationship between each independent variable and the dependent variable.
• The equation allows you to predict the mean value of the dependent variable given the values of the independent variables that you specify.

Linear regression finds the constant and coefficient values for the IVs for a line that best fit your sample data. The graph below shows the best linear fit for the height and weight data points, revealing the mathematical relationship between them. Additionally, you can use the line’s equation to predict future values of the weight given a person’s height.

Linear regression was one of the earliest types of regression analysis to be rigorously studied and widely applied in real-world scenarios. This popularity stems from the relative ease of fitting linear models to data and the straightforward nature of analyzing the statistical properties of these models. Unlike more complex models that relate to their parameters in a non-linear way, linear models simplify both the estimation and the interpretation of data.

In this post, you’ll learn how to interprete linear regression with an example, about the linear formula, how it finds the coefficient estimates , and its assumptions .

Learn more about when you should use regression analysis  and independent and dependent variables .

Linear Regression Example

Suppose we use linear regression to model how the outside temperature in Celsius and Insulation thickness in centimeters, our two independent variables, relate to air conditioning costs in dollars (dependent variable).

Let’s interpret the results for the following multiple linear regression equation:

Air Conditioning Costs$ = 2 * Temperature C – 1.5 * Insulation CM

The coefficient sign for Temperature is positive (+2), which indicates a positive relationship between Temperature and Costs. As the temperature increases, so does air condition costs. More specifically, the coefficient value of 2 indicates that for every 1 C increase, the average air conditioning cost increases by two dollars.

On the other hand, the negative coefficient for insulation (–1.5) represents a negative relationship between insulation and air conditioning costs. As insulation thickness increases, air conditioning costs decrease. For every 1 CM increase, the average air conditioning cost drops by $1.50.

We can also enter values for temperature and insulation into this linear regression equation to predict the mean air conditioning cost.

Learn more about interpreting regression coefficients and using regression to make predictions .

Linear Regression Formula

Linear regression refers to the form of the regression equations these models use. These models follow a particular formula arrangement that requires all terms to be one of the following:

• The constant
• A parameter multiplied by an independent variable (IV)

Then, you build the linear regression formula by adding the terms together. These rules limit the form to just one type:

Dependent variable = constant + parameter * IV + … + parameter * IV

This formula is linear in the parameters. However, despite the name linear regression, it can model curvature. While the formula must be linear in the parameters, you can raise an independent variable by an exponent to model curvature . For example, if you square an independent variable, linear regression can fit a U-shaped curve.

Specifying the correct linear model requires balancing subject-area knowledge, statistical results, and satisfying the assumptions.

Learn more about the difference between linear and nonlinear models and specifying the correct regression model .

How to Find the Linear Regression Line

Linear regression can use various estimation methods to find the best-fitting line. However, analysts use the least squares most frequently because it is the most precise prediction method that doesn’t systematically overestimate or underestimate the correct values when you can satisfy all its assumptions.

The beauty of the least squares method is its simplicity and efficiency. The calculations required to find the best-fitting line are straightforward, making it accessible even for beginners and widely used in various statistical applications. Here’s how it works:

• Objective : Minimize the differences between the observed and the linear regression model’s predicted values . These differences are known as “ residuals ” and represent the errors in the model values.
• Minimizing Errors : This method focuses on making the sum of these squared differences as small as possible.
• Best-Fitting Line : By finding the values of the model parameters that achieve this minimum sum, the least squares method effectively determines the best-fitting line through the data points. 

By employing the least squares method in linear regression and checking the assumptions in the next section, you can ensure that your model is as precise and unbiased as possible. This method’s ability to minimize errors and find the best-fitting line is a valuable asset in statistical analysis.

Assumptions

Linear regression using the least squares method has the following assumptions:

• A linear model satisfactorily fits the relationship.
• The residuals follow a normal distribution.
• The residuals have a constant scatter.
• Independent observations.
• The IVs are not perfectly correlated.

Residuals are the difference between the observed value and the mean value that the model predicts for that observation. If you fail to satisfy the assumptions, the results might not be valid.

Learn more about the assumptions for ordinary least squares and How to Assess Residual Plots .

Yan, Xin (2009),  Linear Regression Analysis: Theory and Computing

Share this:

Reader Interactions

May 9, 2024 at 9:10 am

Why not perform centering or standardization with all linear regression to arrive at a better estimate of the y-intercept?

May 9, 2024 at 4:48 pm

I talk about centering elsewhere. This article just covers the basics of what linear regression does.

A little statistical niggle on centering creating a “better estimate” of the y-intercept. In statistics, there’s a specific meaning to “better estimate,” relating to precision and a lack of bias. Centering (or standardizing) doesn’t create a better estimate in that sense. It can create a more interpretable value in some situations, which is better in common usage.

August 16, 2023 at 5:10 pm

Hi Jim, I’m trying to understand why the Beta and significance changes in a linear regression, when I add another independent variable to the model. I am currently working on a mediation analysis, and as you know the linear regression is part of that. A simple linear regression between the IV (X) and the DV (Y) returns a statistically significant result. But when I add another IV (M), X becomes insignificant. Can you explain this? Seeking some clarity, Peta.

August 16, 2023 at 11:12 pm

This is a common occurrence in linear regression and is crucial for mediation analysis.

By adding M (mediator), it might be capturing some of the variance that was initially attributed to X. If M is a mediator, it means the effect of X on Y is being channeled through M. So when M is included in the model, it’s possible that the direct effect of X on Y becomes weaker or even insignificant, while the indirect effect (through M) becomes significant.

If X and M share variance in predicting Y, when both are in the model, they might “compete” for explaining the variance in Y. This can lead to a situation where the significance of X drops when M is added.

I hope that helps!

July 31, 2022 at 7:56 pm

July 30, 2022 at 2:49 pm

Jim, Hi! I am working on an interpretation of multiple linear regression. I am having a bit of trouble getting help. is there a way to post the table so that I may initiate a coherent discussion on my interpretation?

April 28, 2022 at 3:24 pm

Is it possible that we get significant correlations but no significant prediction in a multiple regression analysis? I am seeing that with my data and I am so confused. Could mediation be a factor (i.e IVs are not predicting the outcome variables because the relationship is made possible through mediators)?

April 29, 2022 at 4:37 pm

I’m not sure what you mean by “significant prediction.” Typically, the predictions you obtain from regression analysis will be a fitted value (the prediction) and a prediction interval that indicates the precision of the prediction (how close is it likely to be to the correct value). We don’t usually refer to “significance” when talking about predictions. Can you explain what you mean? Thanks!

March 25, 2022 at 7:19 am

I want to do a multiple regression analysis is SPSS (creating a predictive model), where IQ is my dependent variable and my independent variables contains of different cognitive domains. The IQ scores are already scaled for age. How can I controlling my independent variables for age, whitout doing it again for the IQ scores? I can’t add age as an independent variable in the model.

I hope that you can give me some advise, thank you so much!

March 28, 2022 at 9:27 pm

If you include age as an independent variable, the model controls for it while calculating the effects of the other IVs. And don’t worry, including age as an IV won’t double count it for IQ because that is your DV.

March 2, 2022 at 8:23 am

Hi Jim, Is there a reason you would want your covariates to be associated with your independent variable before including them in the model? So in deciding which covariates to include in the model, it was specified that covariates associated with both the dependent variable and independent variable at p<0.10 will be included in the model.

My question is why would you want the covariates to be associated with the independent variable?

March 2, 2022 at 4:38 pm

In some cases, it’s absolutely crucial to include covariates that correlate with other independent variables, although it’s not a sufficient reason by itself. When you have a potential independent variable that correlates with other IVs and it also correlates with the dependent variable, it becomes a confounding variable and omitting it from the model can cause a bias in the variables that you do include. In this scenario, the degree of bias depends on the strengths of the correlations involved. Observational studies are particularly susceptible to this type of omitted variable bias. However, when you’re performing a true, randomized experiment, this type of bias becomes a non-issue.

I’ve never heard of a formalized rule such as the one that you mention. Personally, I wouldn’t use p-values to make this determination. You can have low p-values for weak correlation in some cases. Instead, I’d look at the strength of the correlations between IVs. However, it’s not a simple as a single criterial like that. The strength of the correlation between the potential IV and the DV also plays a role.

I’ve written an article about that discusses these issues in more detail, read Confounding Variables Can Bias Your Results .

February 28, 2022 at 8:19 am

Jim, as if by serendipity: having been on your mailing list for years, I looked up your information on multiple regression this weekend for a grad school advanced statistics case study. I’m a fan of your admirable gift to make complicated topics approachable and digestible. Specifically, I was looking for information on how pronounced the triangular/funnel shape must be–and in what directions it may point–to suggest heteroscedasticity in a regression scatterplot of standardized residuals vs standardized predicted values. It seemed to me that my resulting plot of a 5 predictor variable regression model featured an obtuse triangular left point that violated homoscedasticity; my professors disagreed, stating the triangular “funnel” aspect would be more prominent and overt. Thus, should you be looking for a new future discussion point, my query to you then might be some pearls on the nature of a qualifying heteroscedastic funnel shape: How severe must it be? Is there a quantifiable magnitude to said severity, and if so, how would one quantify this and/or what numeric outputs in common statistical software would best support or deny a suspicion based on graphical interpretation? What directions can the funnel point; are only some directions suggestive, whereby others are not? Thanks for entertaining my comment, and, as always, thanks for doing what you do.

Comments and Questions Cancel reply

If you could change one thing about college, what would it be?

Graduate faster

Better quality online classes

Flexible schedule

Access to top-rated instructors

The Complete Guide To Simple Regression Analysis

08.08.2023 • 8 min read

Sarah Thomas

Subject Matter Expert

Learn what simple regression analysis means and why it’s useful for analyzing data, and how to interpret the results.

In This Article

What Is Simple Linear Regression Analysis?

Linear regression equation, how to perform linear regression, linear regression assumptions, how do you find the regression line, how to interpret the results of simple regression.

What is the relationship between parental income and educational attainment or hours spent on social media and anxiety levels? Regression is a versatile statistical tool that can help you answer these types of questions. It’s a tool that lets you model the relationship between two or more variables .

The applications of regression are endless. You can use it as a machine learning algorithm to make predictions. You can use it to establish correlations, and in some cases, you can use it to uncover causal links in your data.

In this article, we’ll tell you everything you need to know about the most basic form of regression analysis: the simple linear regression model.

Simple linear regression is a statistical tool you can use to evaluate correlations between a single independent variable (X) and a single dependent variable (Y). The model fits a straight line to data collected for each variable, and using this line, you can estimate the correlation between X and Y and predict values of Y using values of X.

As a quick example, imagine you want to explore the relationship between weight (X) and height (Y). You collect data from ten randomly selected individuals, and you plot your data on a scatterplot like the one below.

In the scatterplot, each point represents data collected for one of the individuals in your sample. The blue line is your regression line. It models the relationship between weight and height using observed data. Not surprisingly, we see ‌the regression line is upward-sloping, indicating a positive correlation between weight and height. Taller people tend to be heavier than shorter people.

Once you have this line, you can measure how strong the correlation is between height and weight. You can estimate the height of somebody ‌not in your sample by plugging their weight into the regression equation.

The equation for a simple linear regression is:

X is your independent variable

Y is an estimate of your dependent variable

β 0 \beta_0 β 0 ​ is the constant or intercept of the regression line, which is the value of Y when X is equal to zero

β 1 \beta_1 β 1 ​ is the regression coefficient, which is the slope of the regression line and your estimate for the change in Y given a 1-unit change in X

ε \varepsilon ε is the error term of the regression

You may notic‌e the formula for a regression looks very similar to the equation of a line (y=mX+b). That’s because linear regression is a line! It’s a line fitted to data that you can use to estimate the values of one variable using the value of a correlated variable.

You can build a simple linear regression model in 5 steps.

1. Collect data

Collect data for two variables (X and Y). Y is your dependent variable, which is the variable you want to estimate using the regression. X is your independent variable—the variable you use as an input in your regression.

2. Plot the data on a scatter plot

Plot the values of X and Y on a scatter plot with values of X plotted along the horizontal x-axis and values of Y plotted on the vertical y-axis.

3. Calculate a correlation coefficient

Calculate a correlation coefficient to determine the strength of the linear relationship between your two variables.

4. Fit a regression to the data

Find the regression line using the ordinary least-squares method. (You can do this by hand; but it’s much easier to use statistical software like Desmos, Excel, R, or Stata.)

5. Assess the regression line

Once you have the regression line, assess how well your model performs by checking to see how well the model predicts values of Y.

The key assumptions we make when using a simple linear regression model are:

The relationship between X and Y (if it exists) is linear.

Independence

The residuals of your model are independent.

Homoscedasticity

The variance of the residual is constant across values of the independent variable.

The residuals are normally distributed .

You should not use a simple linear regression unless it’s reasonable to make these assumptions.

Simple linear regression involves fitting a straight line to your dataset. We call this line the line of best fit or the regression line. The most common method for finding this line is OLS (or the Ordinary Least Squares Method).

In OLS, we find the regression line by minimizing the sum of squared residuals —also called squared errors. Anytime you draw a straight line through your data, there will be a vertical distance between each ‌point on your scatter plot and the regression line. These vertical distances are called residuals (or errors).

They represent the difference between the actual values of your dependent variable Y i Y_i Y i ​ , and the predicted value of that variable, Y ^ i \widehat{Y}_i Y i ​ . The regression you find with OLS is the line that minimizes the sum of squared residuals.

You can calculate the OLS regression line by hand, but it’s much easier to do so using statistical software like Excel, Desmos, R, or Stata. In this video, Professor AnnMaria De Mars explains how to find the OLS regression equation using Desmos.

Depending on the software you use, the results of your regression analysis may look ‌different. In general, however, your software will display output tables summarizing the main characteristics of your regression.

The values you should be looking for in these output tables fall under three categories:

Coefficients

Regression statistics

This is the β 0 \beta_0 β 0 ​ value in your regression equation. It is the y-intercept of your regression line, and it is the estimate of Y when X is equal to zero.

Next to your intercept, you’ll see columns in the table showing additional information about the intercept. These include a standard error, p-value, T-stat, and confidence interval. You can use these values to test whether the estimate of your intercept is statistically significant .

Regression coefficient

This is the β 1 \beta_1 β 1 ​ of your regression equation. It’s the slope of the regression line, and it tells you how much Y should change in response to a 1-unit change in X.

Similar to the intercept, the regression coefficient will have columns to the right of it. They'll show a standard error, p-value , T-stat, and confidence interval. Use these values to test whether your parameter estimate of β 1 \beta_1 β 1 ​ is statistically significant.

Regression Statistics

Correlation coefficient (or multiple r).

This is the Pearson Correlation coefficient. It measures the strength of the correlation between X and Y.

R-squared (or the coefficient of determination)

We calculate this value by squaring the correlation coefficient. The independent variable can explain how much of the variance in your dependent variable. You can convert R 2 R^2 R 2 into a percentage by multiplying it by 100.

Standard error of the residuals

The standard error of the residuals is the average value of the errors in your model. It is the average vertical distance between each point on your scatter plot and the regression line. We measure this value in the same units as your dependent variable.

Degrees of freedom

In simple linear regression, the degrees of freedom equal the number of data points you used minus the two estimated parameters. The parameters are the intercept and regression coefficient.

Some software will also output a 5-number summary of your residuals. It'll show the minimum, first quartile , median , third quartile, and maximum values of your residuals.

P-value (or Significance F) - This is the p-value of your regression model.

It returns a hypothesis test's results where the null hypothesis is that no relationship exists between X and Y. The alternative hypothesis is that a linear relationship exists between X and Y.

If you are using a significance level (or alpha level) of 0.05, you would reject the null hypothesis if the p-value is less than or equal to 0.05. You would fail to reject the null hypothesis if your p-value is greater than 0.05.

What are correlations?

A correlation is a measure of the relationship between two variables.

Positive Correlations - If two variables, X and Y, have a positive linear correlation, Y tends to increase as X increases, and Y tends to decrease as X decreases. In other words, the two variables tend to move together in the same direction.

Negative Correlations - Two variables, X and Y, have a negative correlation if Y tends to increase as X decreases and Y tends to decrease as X increases. (i.e., The values of the two variables tend to move in opposite directions).

What’s the difference between the dependent and independent variables in a regression?

A simple linear regression involves two variables: X, the input or independent variable, and Y, the output or dependent variable. The independent variable is the variable you want to estimate using the regression. Its estimated value “depends” on the parameters and other variables of the model.

The independent variable—also called the predictor variable—is an input in the model. Its value does not depend on the other elements of the model.

Is the correlation coefficient the same as the regression coefficient?

The correlation coefficient and the regression coefficient will both have the same sign (positive or negative), but they are not the same. The only case where these two values will be equal is when the values of X and Y have been standardized to the same scale.

What is a correlation coefficient?

A correlation coefficient—or Pearson’s correlation coefficient —measures the strength of the linear relationship between X and Y. It’s a number ranging between -1 and 1. The closer a coefficient correlation is to 0, the weaker the correlation is between X and Y.

The closer the correlation coefficient is to 1 or -1, the stronger the correlation. Points on a scatter plot will be more dispersed around the regression line when the correlation between X and Y is weak, and the points will be more tightly clustered around the regression line when the correlation is strong.

What is the regression coefficient?

The regression coefficient, β 1 \beta_1 β 1 ​ , is the slope of the regression line. It provides you with an estimate of how much the dependent variable, Y, will change in response to a 1-unit increase in the dependent variable, X.

The regression coefficient can be any number from − ∞ -\infty − ∞ to ∞ \infty ∞ . A positive regression coefficient implies a positive correlation between X and Y, and a negative regression coefficient implies a negative correlation.

Can I use linear regression in Excel?

Yes. The easiest way to add a simple linear regression line in Excel is to install and use Excel’s “Analysis Toolpak” add-in. To do this, go to Tools > Excel Add-ins and select the “Analysis Toolpak.”

Next, follow these steps.

In your spreadsheet, enter your data for X and Y in two columns

Navigate to the “Data” tab and click on the “Data Analysis” icon

From the list of analysis tools, select “Regression” and click “OK”

Select the data for Y and X respectively where it says “Input Y Range” and “Input X Range”

If you’ve labeled your columns with the names of your X and Y variables, click on the “Labels” checkbox.

You can further customize where you want your regression in your workbook and what additional information you would like Excel to display.

Once you’ve finished customizing, click “OK”

Your regression results will display next to your data or in a new sheet.

Is linear regression used to establish causal relationships?

Correlations are not equivalent to causation. If two variables are correlated, you cannot immediately conclude ‌one causes the other to change. A linear regression will immediately indicate whether two variables correlate. But you’ll need to include more variables in your model and use regression with causal theories to draw conclusions about causal relationships.

What are some other types of regression analysis?

Simple linear regression is the most basic form of regression analysis. It involves ‌one independent variable and one dependent variable. Once you get a handle on this model, you can move on to more sophisticated forms of regression analysis. These include multiple linear regression and nonlinear regression.

Multiple linear regression is a model that estimates the linear relationship between variables using one dependent variable and multiple predictor variables. Nonlinear regression is a method used to estimate nonlinear relationships between variables.

Explore Outlier's Award-Winning For-Credit Courses

Outlier (from the co-founder of MasterClass) has brought together some of the world's best instructors, game designers, and filmmakers to create the future of online college.

Check out these related courses:

Intro to Statistics

How data describes our world.

Intro to Microeconomics

Why small choices have big impact.

Intro to Macroeconomics

How money moves our world.

Intro to Psychology

The science of the mind.

Related Articles

Calculating Logarithmic Regression Step-By-Step

Learn about logarithmic regression and the steps to calculate it. We’ll also break down what a logarithmic function is, why it’s useful, and a few examples.

What Is the Interquartile Range (IQR)?

Learn what the interquartile range is, why it’s used in Statistics and how to calculate it. Also read about how it can be helpful for finding outliers.

Calculate Outlier Formula: A Step-By-Step Guide

This article is an overview of the outlier formula and how to calculate it step by step. It’s also packed with examples and FAQs to help you understand it.

Further Reading

What is statistical significance & why learn it, mean absolute deviation (mad) - meaning & formula, discrete & continuous variables with examples, population vs. sample: the big difference, why is statistics important, how to make a box plot.

Simple linear regression

Simple linear regression #.

Fig. 9 Simple linear regression #

Errors: \(\varepsilon_i \sim N(0,\sigma^2)\quad \text{i.i.d.}\)

Fit: the estimates \(\hat\beta_0\) and \(\hat\beta_1\) are chosen to minimize the (training) residual sum of squares (RSS):

Sample code: advertising data #

Estimates \(\hat\beta_0\) and \(\hat\beta_1\) #.

A little calculus shows that the minimizers of the RSS are:

Assessing the accuracy of \(\hat \beta_0\) and \(\hat\beta_1\) #

Fig. 10 How variable is the regression line? #

Based on our model #

The Standard Errors for the parameters are:

95% confidence intervals:

Hypothesis test #

Null hypothesis \(H_0\) : There is no relationship between \(X\) and \(Y\) .

Alternative hypothesis \(H_a\) : There is some relationship between \(X\) and \(Y\) .

Based on our model: this translates to

\(H_0\) : \(\beta_1=0\) .

\(H_a\) : \(\beta_1\neq 0\) .

Test statistic:

Under the null hypothesis, this has a \(t\) -distribution with \(n-2\) degrees of freedom.

Sample output: advertising data #

Interpreting the hypothesis test #.

If we reject the null hypothesis, can we assume there is an exact linear relationship?

No. A quadratic relationship may be a better fit, for example. This test assumes the simple linear regression model is correct which precludes a quadratic relationship.

If we don’t reject the null hypothesis, can we assume there is no relationship between \(X\) and \(Y\) ?

No. This test is based on the model we posited above and is only powerful against certain monotone alternatives. There could be more complex non-linear relationships.

Save 10% on All AnalystPrep 2024 Study Packages with Coupon Code BLOG10 .

• Payment Plans
• Product List
• Partnerships

• Try Free Trial
• Study Packages
• Levels I, II & III Lifetime Package
• Video Lessons
• Study Notes
• Practice Questions
• Levels II & III Lifetime Package
• About the Exam
• About your Instructor
• Part I Study Packages
• Part I & Part II Lifetime Package
• Part II Study Packages
• Exams P & FM Lifetime Package
• Quantitative Questions
• Verbal Questions
• Data Insight Questions
• Live Tutoring
• About your Instructors
• EA Practice Questions
• Data Sufficiency Questions
• Integrated Reasoning Questions

Hypothesis Testing in Regression Analysis

Hypothesis testing is used to confirm if the estimated regression coefficients bear any statistical significance.  Either the confidence interval approach or the t-test approach can be used in hypothesis testing. In this section, we will explore the t-test approach.

The t-test Approach

The following are the steps followed in the performance of the t-test:

• Set the significance level for the test.
• Formulate the null and the alternative hypotheses.

$$t=\frac{\widehat{b_1}-b_1}{s_{\widehat{b_1}}}$$

\(b_1\) = True slope coefficient.

\(\widehat{b_1}\) = Point estimate for \(b_1\)

\(b_1 s_{\widehat{b_1\ }}\) = Standard error of the regression coefficient.

• Compare the absolute value of the t-statistic to the critical t-value (t_c). Reject the null hypothesis if the absolute value of the t-statistic is greater than the critical t-value i.e., \(t\ >\ +\ t_{critical}\ or\ t\ <\ –t_{\text{critical}}\).

Example: Hypothesis Testing of the Significance of Regression Coefficients

An analyst generates the following output from the regression analysis of inflation on unemployment:

$$\small{\begin{array}{llll}\hline{}& \textbf{Regression Statistics} &{}&{}\\ \hline{}& \text{Multiple R} & 0.8766 &{} \\ {}& \text{R Square} & 0.7684 &{} \\ {}& \text{Adjusted R Square} & 0.7394 & {}\\ {}& \text{Standard Error} & 0.0063 &{}\\ {}& \text{Observations} & 10 &{}\\ \hline {}& & & \\ \hline{} & \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t-Stat}\\ \hline \text{Intercept} & 0.0710 & 0.0094 & 7.5160 \\\text{Forecast (Slope)} & -0.9041 & 0.1755 & -5.1516\\ \hline\end{array}}$$

At the 5% significant level, test the null hypothesis that the slope coefficient is significantly different from one, that is,

$$ H_{0}: b_{1} = 1\ vs. \ H_{a}: b_{1}≠1 $$

The calculated t-statistic, \(\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}\) is equal to:

$$\begin{align*}\text{t}& = \frac{-0.9041-1}{0.1755}\\& = -10.85\end{align*}$$

The critical two-tail t-values from the table with \(n-2=8\) degrees of freedom are:

$$\text{t}_{c}=±2.306$$

Notice that \(|t|>t_{c}\) i.e., (\(10.85>2.306\))

Therefore, we reject the null hypothesis and conclude that the estimated slope coefficient is statistically different from one.

Note that we used the confidence interval approach and arrived at the same conclusion.

Question Neeth Shinu, CFA, is forecasting price elasticity of supply for a certain product. Shinu uses the quantity of the product supplied for the past 5months as the dependent variable and the price per unit of the product as the independent variable. The regression results are shown below. $$\small{\begin{array}{lccccc}\hline \textbf{Regression Statistics} & & & & & \\ \hline \text{Multiple R} & 0.9971 & {}& {}&{}\\ \text{R Square} & 0.9941 & & & \\ \text{Adjusted R Square} & 0.9922 & & & & \\ \text{Standard Error} & 3.6515 & & & \\ \text{Observations} & 5 & & & \\ \hline {}& \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t Stat} & \textbf{P-value}\\ \hline\text{Intercept} & -159 & 10.520 & (15.114) & 0.001\\ \text{Slope} & 0.26 & 0.012 & 22.517 & 0.000\\ \hline\end{array}}$$ Which of the following most likely reports the correct value of the t-statistic for the slope and most accurately evaluates its statistical significance with 95% confidence?     A. \(t=21.67\); slope is significantly different from zero.     B. \(t= 3.18\); slope is significantly different from zero.     C. \(t=22.57\); slope is not significantly different from zero. Solution The correct answer is A . The t-statistic is calculated using the formula: $$\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}$$ Where: \(b_{1}\) = True slope coefficient \(\widehat{b_{1}}\) = Point estimator for \(b_{1}\) \(\widehat{S_{b_{1}}}\) = Standard error of the regression coefficient $$\begin{align*}\text{t}&=\frac{0.26-0}{0.012}\\&=21.67\end{align*}$$ The critical two-tail t-values from the t-table with \(n-2 = 3\) degrees of freedom are: $$t_{c}=±3.18$$ Notice that \(|t|>t_{c}\) (i.e \(21.67>3.18\)). Therefore, the null hypothesis can be rejected. Further, we can conclude that the estimated slope coefficient is statistically different from zero.

Offered by AnalystPrep

Analysis of Variance (ANOVA)

Predicted value of a dependent variable, discrete uniform distribution.

A discrete random variable can assume a finite or countable number of values.... Read More

Using the Standard Normal Distribution ...

Using the standard normal distribution table, we can confirm that a normally distributed... Read More

The Internal Rate of Return

The internal rate of return is the discount rate that sets the present... Read More

Understanding a Parameter, a Statistic ...

A parameter refers to a measure that is used to describe the characteristic... Read More

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

• Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
• Duis aute irure dolor in reprehenderit in voluptate
• Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

Lesson 2: slr model evaluation, overview section  .

This lesson presents two alternative methods for testing whether a linear association exists between the predictor x and the response y in a simple linear regression model:

\(H_{0}\): \(\beta_{1}\) = 0 versus \(H_{A}\): \(\beta_{1}\) ≠ 0.

One is the t -test for the slope while the other is an analysis of variance (ANOVA) F -test .

As you know, one of the primary goals of this course is to be able to translate a research question into a statistical procedure. Here are two examples of research questions and the alternative statistical procedures that could be used to answer them:

• What statistical procedure answers this research question? We could estimate the regression line and then use the t -test to determine if the slope, \(\beta_{1}\), of the population regression line, is 0.
• Alternatively, we could perform an (analysis of variance) F -test.
• What statistical procedure answers this research question? We could estimate the regression line and then use the t -test to see if the slope, \(\beta_{1}\), of the population regression line, is 0.
• Again, we could alternatively perform an (analysis of variance) F -test.

We also learn a way to check for linearity — the "L" in the "LINE" conditions — using the linear lack of fit test . This test requires replicates, that is multiple observations of y for at least one (preferably more) values of x , and concerns the following hypotheses:

• \(H_{0}\): There is no lack of linear fit.
• \(H_{A}\): There is a lack of linear fit.
• Calculate confidence intervals and conduct hypothesis tests for the population intercept \(\beta_{0}\) and population slope \(\beta_{1}\) using Minitab's regression analysis output.
• Draw research conclusions about the population intercept \(\beta_{0}\) and population slope \(\beta_{1}\) using the above confidence intervals and hypothesis tests.
• Know the six possible outcomes about the slope \(\beta_{1}\) whenever we test whether there is a linear relationship between a predictor x and a response y .
• Understand the "derivation" of the analysis of variance F -test for testing \(H_{0}\): \(\beta_{1} = 0\). That is, understand how the total variation in a response y is broken down into two parts — a component that is due to the predictor x and a component that is just due to random error. And, understand how the expected mean squares tell us to use the ratio MSR/MSE to conduct the test.
• Know how each element of the analysis of variance table is calculated.
• Know what scientific questions can be answered with the analysis of variance F -test.
• Conduct the analysis of variance F -test to test \(H_{0}\): \(\beta_{1} = 0\) versus \(H_{A}\): \(\beta_{1} ≠ 0\).
• Know the similarities and distinctions of the t -test and F -test for testing \(H_{0}\):\(\beta_{1} = 0\).
• Know the t -test for testing that \(\beta_{1}\) = 0, the F -test for testing that \(\beta_{1}\) = 0, and the t-test for testing that \(\rho = 0\) yield similar results, but understand when it makes sense to report the results of each one.
• Calculate all of the values in the lack of fit analysis of variance table.
• Conduct the F -test for lack of fit.
• Know that the (linear) lack of fit test only gives you evidence against linearity. If you reject the null and conclude a lack of linear fit, it doesn't tell you what (non-linear) regression function would work.
• Understand the "derivation" of the linear lack of fit test. That is, understand the decomposition of the error sum of squares, and how the expected mean squares tell us to use the ratio MSLF/MSPE to test for lack of linear fit.

Lesson 2 Code Files Section  

Below is a zip file that contains all the data sets used in this lesson:

STAT501_Lesson02.zip

• couplesheight.txt
• handheight.txt
• heightgpa.txt
• husbandwife.txt
• leadcord.txt
• mens200m.txt
• newaccounts.txt
• signdist.txt
• skincancer.txt
• solutions_conc.txt
• whitespruce.txt