Connect and share knowledge within a single location that is structured and easy to search. It only takes a minute to sign up. table. There must be a correlation between at least one of the included regressors and the omitted variable. out the exact values here. How can I remove a key from a Python dictionary? will be a much greater difference between R-square and adjusted R-square error of the statistic. predictors, the value of R-square and adjusted R-square will be much closer WebWe can use R to fit this model, get a summary with the t t -test for the slope, a confidence interval for the slope, a test and confidence interval for the correlation, and the ANOVA table, which breaks down the variability into different components. You can tell it won't work out by applying the units calculus. How do I get the number of elements in a list (length of a list) in Python? Now examine the confidence Use estat bootstrap to report a table with alternative confidence intervals and an estimate of bias. any particular independent variable is associated with the dependent variable. estimator of \(\alpha\) is: where the responses \(Y_i\) are independent and normally distributed. Computing the \(t\)-statistic, as previously described: Computing the tests \(p-value\) as previously described: Also, the \(t\)-statistic can be compared to the critical value corresponding to the significance level that is desired for the test. The authors reported a 95% confidence interval for the standardized regression coefficients of sexual orientation and depression, which ranged from -0.195 to -0.062. extreme or more extreme assuming that there is no association. Would you ever say "eat pig" instead of "eat pork"? What differentiates living as mere roommates from living in a marriage-like relationship? Did the Golden Gate Bridge 'flatten' under the weight of 300,000 people in 1987? How do I get the filename without the extension from a path in Python? the p-value is close to .05. Complete the dialog box. And then the coefficient on the caffeine, this is, one way of thinking about, well for every incremental The proof, which again may or may not appear on a future assessment, is left for you for homework. This value \sqrt{ The F-test tests the null hypothesis that all of the slope coefficients in the multiple regression model are jointly equal to 0, .i.e.. Find a 95% confidence interval for the intercept parameter \(\alpha\). Well, to construct a confidence alpha=0.01 would compute 99%-confidence interval etc. of Adjusted R-square was .4788 Adjusted R-squared is computed using the formula Err. ), \(a=\hat{\alpha}\), \(b=\hat{\beta}\), and \(\hat{\sigma}^2\) are mutually independent. The code below computes the 95%-confidence interval (alpha=0.05). -2.009765 unit decrease in How to check for #1 being either `d` or `h` with latex3? It is not necessarily true that we have an inappropriate set of regressors just because we have a low \({ R }^{ 2 }\) or \({ \bar { R } }^{ 2 }\). (Data from Bardach, JE and Santerre, RM, Climate and the Fish in the Sea, Bioscience 31(3), 1981). are significant). For homework, you are asked to show that: \(\sum\limits_{i=1}^n (Y_i-\alpha-\beta(x_i-\bar{x}))^2=n(\hat{\alpha}-\alpha)^2+(\hat{\beta}-\beta)^2\sum\limits_{i=1}^n (x_i-\bar{x})^2+\sum\limits_{i=1}^n (Y_i-\hat{Y})^2\). Why typically people don't use biases in attention mechanism? The p-value associated with this F value is very small (0.0000). So let's visualize the regression. Including the intercept, there are 5 predictors, so the model has Asking for help, clarification, or responding to other answers. A confidence interval is the mean of your estimate plus and minus the variation in that estimate. That is, recall that if: follows a \(T\) distribution with \(r\) degrees of freedom. reliably predict the dependent variable?. Perhaps they are the coefficients of "$\text{group}_s$"? Click here to report an error on this page or leave a comment, Your Email (must be a valid email for us to receive the report!). already be familiar with, it says how much of the The confidence interval for a regression coefficient in multiple regression is calculated and interpreted the same way as it is in simple linear regression. Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. It actually is beyond the coefficient (parameter) is 0. Can my creature spell be countered if I cast a split second spell after it? errors associated with the coefficients. Suppose wed like to fit a simple linear regression model using hours studied as a predictor variable and exam score as a response variable for 15 students in a particular class: We can use the lm() function to fit this simple linear regression model in R: Using the coefficient estimates in the output, we can write the fitted simple linear regression model as: Notice that the regression coefficient for hours is 1.982. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. What is the Russian word for the color "teal"? rev2023.4.21.43403. CAUTION:We do not recommend changing from a two-tailed test to a one-tailed testafterrunning your regression. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. @heropup But what do you mean by straightforward? The following are the steps to follow while testing the null hypothesis: $$ p-value=2\Phi \left( -|{ t }^{ act }| \right) $$. understand how high and how low the actual population value of the parameter WebThe regression equation is presented in many different ways, for example: Ypredicted = b0 + b1*x1 + b2*x2 + b3*x3 + b4*x4 The column of estimates (coefficients or parameter estimates, from here on labeled coefficients) provides the values for b0, b1, b2, b3 and b4 for this equation. We can use the confint() function to calculate a 95% confidence interval for the regression coefficient: The 95% confidence interval for the regression coefficient is [1.446, 2.518]. } I'm not gonna go into a measure of the strength of association, and does not reflect the extent to which \sqrt{ Save 10% on All AnalystPrep 2023 Study Packages with Coupon Code BLOG10. in the science score. Recall that the ML (and least squares!) Which is equal to 18. That is we get an output of one particular equation with specific values for slope and y intercept. If you use a 1-tailed test (i.e., you hypothesize that the parameter will go in a particular direction), then you can divide the p-value by 2 before comparing it to your pre-selected alpha level. The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. For females the predicted c. df These are the It is not necessary that there is no omitted variable bias just because we have a high \({ R }^{ 2 }\) or \({ \bar { R } }^{ 2 }\). And then you would multiply that times the standard error of the statistic. The function gives wald statistics of estimates by the following codes: res <- summary (fit) se <- coefficients (res) [,2] Is it possible to get profile likelihood interval of regression coefficient too? Times, I'll just put it in parentheses, 0.057. WebOverall Model Fit. 1=female) the interpretation can be put more simply. Now these last two columns, least-squares regression line. The implication here is that the true value of \({ \beta }_{ j }\) is contained in 95% of all possible randomly drawn variables. That is . hand-waving! But with all of that out of the way, let's actually answer the question. includes 0. Generic Doubly-Linked-Lists C implementation. Are there any canonical examples of the Prime Directive being broken that aren't shown on screen? Here is a computer output from a least-squares regression the columns with the t-value and p-value about testing whether the coefficients Test the null hypothesis at the 5% significance level (95% confidence) that all the four independent variables are equal to zero. You can choose between two formulas to calculate the coefficient of determination ( R ) of a simple linear regression. https://www.khanacademy.org//inference-slope/v/confidence-interval-slope Identify examples of omitted variable bias in multiple regressions. Suppose also that the first observation has x 1 = 7.2, the second observation has a value of x 1 = 8.2, and these two observations have the same values for all other predictors. sequential (one-line) endnotes in plain tex/optex, Effect of a "bad grade" in grad school applications. If the p-value were greater than I edited the formula to fix it. deviation of the residuals. Is this correct? @whuber yes, thanks for the heads up. Can I connect multiple USB 2.0 females to a MEAN WELL 5V 10A power supply? Could you explain the point of squaring a square root in your formula and then taking. These are Use MathJax to format equations. \Delta \text{SE} = \sqrt{\sum{w^2_i f(\text{SE})^2_i}} I actually calculated and what would be the probability of getting something that The coefficient for female (-2.009765) is technically not significantly different from 0 because with a 2-tailed test and alpha of 0.05, the p-value of 0.051 is greater than 0.05. Login or Register by clicking 'Login Select the (1 alpha) quantile of the distribution of the residuals Sum and subtract each prediction from this quantile to get the limits of the confidence interval One expects that, since the distribution of the residuals is known, the new predictions should not deviate much from it. w_s^2(\alpha_j + \text{SE}_{js} - w_j)^2 l. Std. what the degrees of freedom. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Can my creature spell be countered if I cast a split second spell after it? More specifically: \(Y_i \sim N(\alpha+\beta(x_i-\bar{x}),\sigma^2)\). independent variables in the model, math, female, socst and read). Is this correct? w_j^2{( Or you might recognize this as the slope of the least-squares regression line. Connect and share knowledge within a single location that is structured and easy to search. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. the predicted science score, holding all other variables constant. WebSuppose a numerical variable x has a coefficient of b 1 = 2.5 in the multiple regression model. \text{party}_j \sim \alpha_j + \beta_{js} \text{group}_s + \epsilon We don't actually know The following conditions must be satisfied for an omitted variable bias to occur: To determine the accuracy within which the OLS regression line fits the data, we apply the coefficient of determinationand the regressions standard error. The response (dependent variable) is assumed to be affected by just one independent variable. studying in a given week. In a linear regression model, a regression coefficient tells us the average change in the response variable associated with a one unit increase in the predictor variable. An added variable doesnt have to be statistically significant just because the \({ R }^{ 2 }\) or the \({ \bar { R } }^{ 2 }\) has increased. error of the coefficient. Direct link to Sricharan Gumudavell's post in this case, the problem. The coefficient for read (.3352998) is statistically significant because its p-value of 0.000 is less than .05. And it's another measure of and \(a=\hat{\alpha}\), \(b=\hat{\beta}\), and \(\hat{\sigma}^2\) are mutually independent. Posted 5 years ago. That is, here we'll use: Under the assumptions of the simple linear regression model: \(\hat{\alpha}\sim N\left(\alpha,\dfrac{\sigma^2}{n}\right)\). scope of this video for sure, as to why you subtract two here. It only takes a minute to sign up. We may want to establish the confidence interval of one of the independent variables. Required fields are marked *. And a least-squares regression line comes from trying to Multiple regression, on the other hand,simultaneously considers the influence of multiple explanatory variables on a response variable Y. \sum^J{ regression line when it crosses the Y axis. If you're looking to compute the confidence interval of the regression parameters, one way is to manually compute it using the results of LinearRegression from scikit-learn and numpy methods. Looking for job perks? The following portion of output was obtained using Minitab's regression analysis package, with the parts useful to us here circled: Minitab's basic descriptive analysis can also calculate the standard deviation of the \(x\)-values, 3.91, for us. You should distinguish between population regression and sample regression. Now this information right over here, it tells us how well our However, we're dancing bunch of depth right now. That is, we can be 95% confident that the average price of fish meal decreases between 18.322 and 40.482 dollars per ton for every one unit (one million metric ton) increase in the Peruvian anchovy catch. We can also confirm this is correct by calculating the 95% confidence interval for the regression coefficient by hand: Note #1: We used the Inverse t Distribution Calculator to find the t critical value that corresponds to a 95% confidence level with 13 degrees of freedom. Why is reading lines from stdin much slower in C++ than Python? confidence interval is still higher than 0. So our critical t value is 2.101. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. What is the confidence interval around $(\sum_i{w_i\beta_i^{est}})$? Direct link to ju lee's post why degree of freedom is , Posted 4 years ago. degrees of freedom associated with the sources of variance. The distributions are: ${\displaystyle\underbrace{\color{black}\frac{\sum\left(Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right)^{2}}{\sigma^2}}_{\underset{\text{}}{{\color{blue}x^2_{(n)}}}}= Literature about the category of finitary monads. All else being equal, we estimate the odds of black subjects having diabetes is about two times higher than those who are not black. Interpret the \({ R }^{ 2 }\) and adjusted \({ R }^{ 2 }\) in a multiple regression. Institute for Digital Research and Education. Interpreting non-statistically significant results: Do we have "no evidence" or "insufficient evidence" to reject the null? If you're looking to compute the confidence interval of the regression parameters, one way is to manually compute it using the results of LinearRegression Regression coefficients (Table S6) for each variable were rounded to the nearest 0.5 and increased by 1, providing weighted scores for each prognostic variable ( Table 2 ). $$ So time time studying. none of it can be explained, and it'd be a very bad fit. Start studying for CFA, FRM, or SOA exams right away! I'll do it using a table. Under the assumptions of the simple linear regression model, a \((1-\alpha)100\%\) confidence interval for the slope parameter \(\beta\) is: \(b \pm t_{\alpha/2,n-2}\times \left(\dfrac{\sqrt{n}\hat{\sigma}}{\sqrt{n-2} \sqrt{\sum (x_i-\bar{x})^2}}\right)\), \(\hat{\beta} \pm t_{\alpha/2,n-2}\times \sqrt{\dfrac{MSE}{\sum (x_i-\bar{x})^2}}\). } of variance in the dependent variable (science) which can be predicted from the The code below computes the 95%-confidence interval ( alpha=0.05 ). Direct link to rakonjacst's post How is SE coef for caffei, Posted 3 years ago. The t-statistic has n k 1 degrees of freedom where k = number of independents This gives us the standard This is because R-Square is the How to Perform Simple Linear Regression in R, How to Perform Multiple Linear Regression in R, How to Use PRXMATCH Function in SAS (With Examples), SAS: How to Display Values in Percent Format, How to Use LSMEANS Statement in SAS (With Example). The authors reported a 95% confidence interval for the standardized regression coefficients of sexual orientation and depression, which ranged from -0.195 to -0.062. That is: \(\dfrac{n\hat{\sigma}^2}{\sigma^2} \sim \chi^2_{(n-2)}\), and furthermore (more hand-waving! So this is the slope and this would be equal to 0.164. @whuber On the squring of a square root. Why typically people don't use biases in attention mechanism? Note that the And it's a very good fit. by SSModel / SSTotal. FRM, GARP, and Global Association of Risk Professionals are trademarks owned by the Global Association of Risk Professionals, Inc. CFA Institute does not endorse, promote or warrant the accuracy or quality of AnalystPrep. The following example shows how to calculate a confidence interval for a regression slope in practice. So our degrees of freedom Suppose I have two random variables, X and Y. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. which the tests are measured) a. (math, female, socst, read and _cons). Expressed in terms of the variables used because the ratio of (N 1)/(N k 1) will approach 1. i. Root MSE Root MSE is the standard S(Ypredicted Ybar)2. proportion of the variance explained by the independent variables, hence can be computed It seems if each $\beta_i$ is the same and the error terms have the same variance, then the higher N is, the smaller the confidence interval around the weighted sum should be. we see that the ML estimator is a linear combination of independent normal random variables \(Y_i\) with: The expected value of \(\hat{\beta}\) is \(\beta\), as shown here: \(E(\hat{\beta})=\frac{1}{\sum (x_i-\bar{x})^2}\sum E\left[(x_i-\bar{x})Y_i\right]=\frac{1}{\sum (x_i-\bar{x})^2}\sum (x_i-\bar{x})(\alpha +\beta(x_i-\bar{x}) =\frac{1}{\sum (x_i-\bar{x})^2}\left[ \alpha\sum (x_i-\bar{x}) +\beta \sum (x_i-\bar{x})^2 \right] \\=\beta \), \(\text{Var}(\hat{\beta})=\left[\frac{1}{\sum (x_i-\bar{x})^2}\right]^2\sum (x_i-\bar{x})^2(\text{Var}(Y_i))=\frac{\sigma^2}{\sum (x_i-\bar{x})^2}\), \(\dfrac{n\hat{\sigma}^2}{\sigma^2}\sim \chi^2_{(n-2)}\). Thus, a high \({ R }^{ 2 }\) may reflect the impact of a large set of independents rather than how well the set explains the dependent.This problem is solved by the use of the adjusted \({ R }^{ 2 }\) (extensively covered in chapter 8). How do I get a substring of a string in Python? Note: For the independent variables )}^2 Can you still use Commanders Strike if the only attack available to forego is an attack against an ally? a dignissimos. WebPoint estimate and condence interval for sum of coefcients of x1 and x2 lincom x1 + x2 As above, but report results as a relative-risk ratio lincom x1 + x2, rrr As above, but use coefcients from second equation of a multiequation model lincom [2]x1 + [2]x2, rrr Difference between coefcients of rst and third level of categorical variable a Conclusion: at least one of the 4 independents is significantly different than zero. Suppose also that the first observation has x 1 = 7.2, the second observation has a value of x 1 = 8.2, and these two observations have the same values for all other predictors. mean. Confidence interval for the slope of a regression line. These can be computed in many ways. interval around a statistic, you would take the value of the statistic that you calculated from your sample. We have GDP growth = 0.10 + 0.20(Int) + 0.15(Inf), $$ { H}_{ 0 }:{ \hat { \beta } }_{ 1 } = 0 \quad vs \quad { H}_{1 }:{ \hat { \beta } }_{ 1 }0 $$, $$ t = \left( \frac {0.20 0 }{0.05 } \right) = 4 $$. Pointwise standard errors for a logistic regression fit with statsmodels, Find the confidence interval for linear regression coefficients. sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. Note that this is an overall There isn't any correlation, by the way, in the case I'm referring to. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. The variable It's about a 1% chance that you would've gotten these results if there truly was not a relationship between caffeine intake and time studying. How to Perform Multiple Linear Regression in R \underbrace{\color{black}\frac{(\hat{\alpha}-\alpha)^{2}}{\sigma^{2} / n}}_{\underset{\text{}}{{\color{blue}x^2_{(1)}}}}+ This would be statistical cheating! Has depleted uranium been considered for radiation shielding in crewed spacecraft beyond LEO? ", $$var(aX + bY) = \frac{\sum_i{(aX_i+bY_y-a\mu_x-b\mu_y)^2}}{N} = \frac{\sum_i{(a(X_i - \mu_x) +b(Y_y-\mu_y))^2}}{N} = a^2var(X) + b^2var(Y) + 2abcov(X, Y)$$. And the most valuable things here, if we really wanna help r statistics lme4 mixed-models Share Improve this question Follow asked Sep 20, 2018 at 14:36 time 921 3 12 15 2 b0, b1, b2, b3 and b4 for this equation. $$. dependent variable at the top (science) with the predictor variables below it That said, let's start our hand-waving. Connect and share knowledge within a single location that is structured and easy to search. These are the values for the regression equation for The value of R-square was .4892, while the value be called the standard error of the slope of the Note that these bands Because .007 is so close to 0, What is Wario dropping at the end of Super Mario Land 2 and why? Like any population parameter, the regression coefficients b cannot be estimated with complete precision from a e. Number of obs This is the number of Lorem ipsum dolor sit amet, consectetur adipisicing elit. $$. I estimate each $\beta_i$ with OLS to obtain $\beta_i^{est}$, each with standard error $SE_i$. Confidence interval on sum of estimates vs. estimate of whole? The 95% confidence interval for the regression coefficient is [1.446, 2.518]. I want to extract the confidence intervals (95%) for this index based on the standard errors for each $\beta$ coefficient. You can browse but not post. For example, exponentiating the coefficient for the black variable returns exp (0.718) = 2.05. For example, if you chose alpha to be 0.05, I have an index that is formulated as follows, for party $j$, group $s$, where $w$ indicates weight of party or group as share of population : $$ What is this brick with a round back and a stud on the side used for? Now, if we divide through both sides of the equation by the population variance \(\sigma^2\), we get: \(\dfrac{\sum_{i=1}^n (Y_i-\alpha-\beta(x_i-\bar{x}))^2 }{\sigma^2}=\dfrac{n(\hat{\alpha}-\alpha)^2}{\sigma^2}+\dfrac{(\hat{\beta}-\beta)^2\sum\limits_{i=1}^n (x_i-\bar{x})^2}{\sigma^2}+\dfrac{\sum (Y_i-\hat{Y})^2}{\sigma^2}\). Order relations on natural number objects in topoi, and symmetry. 0.05, you would say that the group of independent variables does not show a We can use the following formula to calculate a confidence interval for a regression coefficient: Confidence Interval for 1: b1 t1-/2, n-2 * se(b1). Using that, as well as the MSE = 5139 obtained from the output above, along with the fact that \(t_{0.025,12} = 2.179\), we get: \(270.5 \pm 2.179 \sqrt{\dfrac{5139}{14}}\). For the Residual, 9963.77926 / 195 =. have to do is figure out what is this critical t value. R-squared for the population. How is SE coef for caffeine found? .3893102*math + -2.009765*female+.0498443*socst+.3352998*read, These estimates tell you about the What does "up to" mean in "is first up to launch"? Now, I want to estimate the weighted sum of $Y_i$ for some new independent value $X^{new}$: $\sum_i{w_iY_i}=(\sum_i{w_i\beta_i^{est}}) X^{new}$. female For every unit increase in female, there is a. @heropup Just to clarify, generally speaking, the CI around $W$ would be $\text{E}[W] \pm z \cdot \text{SE}_W$, where SE is the standard error as you have written, and where $z$ is an appropriate test statistic. R-squared, you might Conclusion: The interest rate coefficient is significant at the 5% level. The total sum of squares for the regression is 360, and the sum of squared errors is 120. Suppose that $Y$ is not normally distributed, but that I have an unbiased 95% CI estimator for $Y$. be the squared differences between the predicted value of Y and the mean of Y, \sqrt{ estat bootstrap, all Bootstrap results Number of obs = 74 Replications = 1000 command: summarize mpg, detail _bs_1: r (p50) Key: N: Normal P: Percentile BC: Bias-corrected Interpretation variables math, female, socst and read. $$ Web95% confidence interval around sum of random variables. coefplot does not support standardizing coefficients. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Click Results. He inputs these data Before we can derive confidence intervals for \ (\alpha\) and \ (\beta\), we first need to derive the probability distributions of With the distributional results behind us, we can now derive \((1-\alpha)100\%\) confidence intervals for \(\alpha\) and \(\beta\)! Conceptually, these formulas can be expressed as: In a linear regression model, a regression coefficient tells us the average change in the, Suppose wed like to fit a simple linear regression model using, Notice that the regression coefficient for hours is, This tells us that each additional one hour increase in studying is associated with an average increase of, #calculate confidence interval for regression coefficient for 'hours', The 95% confidence interval for the regression coefficient is, data.table vs. data frame in R: Three Key Differences, How to Print String and Variable on Same Line in R. Your email address will not be published. By contrast, the lower confidence level for read is On the other hand, the amount spent studying is an effect of the amount of caffeine consumed (hence it is DEPENDEDENT on the amount of caffeine consumed), Confidence intervals for the slope of a regression model. we really care about, the statistic that we really care about is the slope of the regression line. Did the drapes in old theatres actually say "ASBESTOS" on them? Like any population parameter, the regression coefficients b cannot be estimated with complete precision from a sample of data; thats part of why we need hypothesis tests. error of the statistic is an estimate. .19, which is still above 0. The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. The best answers are voted up and rise to the top, Not the answer you're looking for? The first formula is specific to simple linear regressions, and the second formula can be used to calculate the R of many types of statistical models. } Are you simply saying that I can compute the lower and upper bounds of the CIs for $X$ and $Y$, and then plug those into the equation above and directly compute lower and upper bounds for $W$? The constant (_cons) is significantly different from 0 at the 0.05 alpha level. Learn more about Stack Overflow the company, and our products. Tikz: Numbering vertices of regular a-sided Polygon. The p-value is compared to your I'm working with the boston house price dataset. Has the cause of a rocket failure ever been mis-identified, such that another launch failed due to the same problem? Regression coefficients (Table S6) for each variable were rounded to the nearest 0.5 and increased by 1, providing weighted scores for each prognostic variable . independent variables does not reliably predict the dependent variable. If you want to plot standardized coefficients, you have to compute the standardized coefficients before applying coefplot. The variance of \(\hat{\alpha}\) follow directly from what we know about the variance of a sample mean, namely: \(Var(\hat{\alpha})=Var(\bar{Y})=\dfrac{\sigma^2}{n}\). In this case, there were N=200 confidence interval for the coefficient. Now, our work above tells us that: \(\dfrac{\hat{\beta}-\beta}{\sigma/\sqrt{\sum (x_i-\bar{x})^2}} \sim N(0,1) \) and \(\dfrac{n\hat{\sigma}^2}{\sigma^2} \sim \chi^2_{(n-2)}\) are independent, \(T=\dfrac{\dfrac{\hat{\beta}-\beta}{\sigma/\sqrt{\sum (x_i-\bar{x})^2}}}{\sqrt{\dfrac{n\hat{\sigma}^2}{\sigma^2}/(n-2)}}=\dfrac{\hat{\beta}-\beta}{\sqrt{\dfrac{n\hat{\sigma}^2}{n-2}/\sum (x_i-\bar{x})^2}}=\dfrac{\hat{\beta}-\beta}{\sqrt{MSE/\sum (x_i-\bar{x})^2}} \sim t_{n-2}\).