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In A simple linear regression, $SSR/\sigma^2$ follows chi square with one degree of freedom where SSR denotes the sum of squares due to regression.

Although I get that it will have one degree of freedom, I just am not able to think how to prove that $SSR/\sigma^2$ will follow chi square. How do I prove that it's the square of a standard normal?. I am new to statistics and self learning. I tried searching the site. But most of the answers are given in the context of multi linear regression. I don't understand multi linear much right now . I just need a hint in proving that it can be expressed as the square of a standard normal...

Could someone please help. Thanks a lot

MathMan
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  • Please search our site. – whuber Mar 10 '21 at 19:35
  • @whuber I tried searching. But most of the answers are given in the context of multi linear regression. I don't understand multi linear much right now – MathMan Mar 10 '21 at 19:38
  • OK, I found more answers by searching "anova degrees freedom." Some interesting related posts can be found by searching for "cochran's theorem." – whuber Mar 10 '21 at 19:42
  • @whuber I'm sorry but I understand the degree of freedom part. I just need help in proving that it follows chi square in a simple linear regression. Could you please guide me. I have been stuck for long – MathMan Mar 10 '21 at 19:46
  • That it must be a multiple of a chi-squared distribution follows from expressing it as the square of a Normally distributed variable. You can search for many explanations of this by including "normal chi squared distribution" in your search. – whuber Mar 10 '21 at 19:48
  • @whuber will the chi square feature be also be proved using Cochran theorem? – MathMan Mar 10 '21 at 19:48
  • Cochran's Theorem shows how to express SSR as the square of a variable. The rest requires only knowing that the chi-squared distribution is defined as being that of the square of a standard Normal variable. – whuber Mar 10 '21 at 19:49
  • @whuber aah I see. So, I will have to know Cochran's theorem to be able to see that it can be expressed as the square of a standard normal.... – MathMan Mar 10 '21 at 19:52
  • @whuber the theorem involves a lot of matrix theory. Do you think a simpler proof is possible in the case of simple linear regression? – MathMan Mar 10 '21 at 19:58

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