Find a test for $H_{0} : sigma_{1}^{2} ne sigma_{2}^{2}$, against $H_{1} : sigma_{1}^{2} =sigma_{2}^{2}$












0












$begingroup$


Consider $X_{1}dots X_{n} $ ~ $N(a_{1},sigma_{1}^{2})$ and $Y_{1}dots Y_{m} $ ~ $N(a_{2},sigma_{2}^{2})$, and they are independent. We need to find a criteria for $H_{0}:: sigma_{1}^{2} ne sigma_{2}^{2}$.



First of all, let's consider (if $H_{0}$ is true) $frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}$ distributed as $N(0,sigma^{2}frac{a_{2}^2 +a_{1}^2}{a_{1}^2a_{2}^2})$, then after considering of $dfrac{frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}}{sigmasqrt{frac{a_1^2 +a_2^2}{a_1^2 a_2^2}}}$. Now we need to estimate $sigma$ as $S^{2}$, after simplifying we have :
$dfrac{bar{X}sqrt{n}a_{2} - bar{Y}sqrt{m}a_{1}}{sqrt{a_2^2+a_1^2}S} >t_{1-alpha /2}$ is a Student-test.



Am I right?






probability-theory statistics







share|cite|improve this question











$endgroup$












  • $begingroup$
    You don't have to use the student's t-distribution unless your sample is small.
    $endgroup$
    – Frpzzd
    Dec 11 '18 at 17:07










  • $begingroup$
    @Frpzzd actually I'm interested in the correctness of my proof
    $endgroup$
    – openspace
    Dec 11 '18 at 17:18












  • $begingroup$
    What do you mean by "criteria for $H_0$..."? Are you deriving a test for testing $H_0$ against some $H_1$?
    $endgroup$
    – StubbornAtom
    Dec 11 '18 at 17:28










  • $begingroup$
    @StubbornAtom yes, it's right to call it a test
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29










  • $begingroup$
    @StubbornAtom added
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29
















0












$begingroup$


Consider $X_{1}dots X_{n} $ ~ $N(a_{1},sigma_{1}^{2})$ and $Y_{1}dots Y_{m} $ ~ $N(a_{2},sigma_{2}^{2})$, and they are independent. We need to find a criteria for $H_{0}:: sigma_{1}^{2} ne sigma_{2}^{2}$.



First of all, let's consider (if $H_{0}$ is true) $frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}$ distributed as $N(0,sigma^{2}frac{a_{2}^2 +a_{1}^2}{a_{1}^2a_{2}^2})$, then after considering of $dfrac{frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}}{sigmasqrt{frac{a_1^2 +a_2^2}{a_1^2 a_2^2}}}$. Now we need to estimate $sigma$ as $S^{2}$, after simplifying we have :
$dfrac{bar{X}sqrt{n}a_{2} - bar{Y}sqrt{m}a_{1}}{sqrt{a_2^2+a_1^2}S} >t_{1-alpha /2}$ is a Student-test.



Am I right?






probability-theory statistics







share|cite|improve this question











$endgroup$












  • $begingroup$
    You don't have to use the student's t-distribution unless your sample is small.
    $endgroup$
    – Frpzzd
    Dec 11 '18 at 17:07










  • $begingroup$
    @Frpzzd actually I'm interested in the correctness of my proof
    $endgroup$
    – openspace
    Dec 11 '18 at 17:18












  • $begingroup$
    What do you mean by "criteria for $H_0$..."? Are you deriving a test for testing $H_0$ against some $H_1$?
    $endgroup$
    – StubbornAtom
    Dec 11 '18 at 17:28










  • $begingroup$
    @StubbornAtom yes, it's right to call it a test
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29










  • $begingroup$
    @StubbornAtom added
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29














0












0








0





$begingroup$


Consider $X_{1}dots X_{n} $ ~ $N(a_{1},sigma_{1}^{2})$ and $Y_{1}dots Y_{m} $ ~ $N(a_{2},sigma_{2}^{2})$, and they are independent. We need to find a criteria for $H_{0}:: sigma_{1}^{2} ne sigma_{2}^{2}$.



First of all, let's consider (if $H_{0}$ is true) $frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}$ distributed as $N(0,sigma^{2}frac{a_{2}^2 +a_{1}^2}{a_{1}^2a_{2}^2})$, then after considering of $dfrac{frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}}{sigmasqrt{frac{a_1^2 +a_2^2}{a_1^2 a_2^2}}}$. Now we need to estimate $sigma$ as $S^{2}$, after simplifying we have :
$dfrac{bar{X}sqrt{n}a_{2} - bar{Y}sqrt{m}a_{1}}{sqrt{a_2^2+a_1^2}S} >t_{1-alpha /2}$ is a Student-test.



Am I right?






probability-theory statistics







share|cite|improve this question











$endgroup$




Consider $X_{1}dots X_{n} $ ~ $N(a_{1},sigma_{1}^{2})$ and $Y_{1}dots Y_{m} $ ~ $N(a_{2},sigma_{2}^{2})$, and they are independent. We need to find a criteria for $H_{0}:: sigma_{1}^{2} ne sigma_{2}^{2}$.



First of all, let's consider (if $H_{0}$ is true) $frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}$ distributed as $N(0,sigma^{2}frac{a_{2}^2 +a_{1}^2}{a_{1}^2a_{2}^2})$, then after considering of $dfrac{frac{bar{X}sqrt{n}}{a_{1}} - frac{bar{Y}sqrt{m}}{a_{2}}}{sigmasqrt{frac{a_1^2 +a_2^2}{a_1^2 a_2^2}}}$. Now we need to estimate $sigma$ as $S^{2}$, after simplifying we have :
$dfrac{bar{X}sqrt{n}a_{2} - bar{Y}sqrt{m}a_{1}}{sqrt{a_2^2+a_1^2}S} >t_{1-alpha /2}$ is a Student-test.



Am I right?






probability-theory statistics




probability-theory statistics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 11 '18 at 17:53







openspace

















asked Dec 11 '18 at 16:58









openspaceopenspace

3,4192822




3,4192822












  • $begingroup$
    You don't have to use the student's t-distribution unless your sample is small.
    $endgroup$
    – Frpzzd
    Dec 11 '18 at 17:07










  • $begingroup$
    @Frpzzd actually I'm interested in the correctness of my proof
    $endgroup$
    – openspace
    Dec 11 '18 at 17:18












  • $begingroup$
    What do you mean by "criteria for $H_0$..."? Are you deriving a test for testing $H_0$ against some $H_1$?
    $endgroup$
    – StubbornAtom
    Dec 11 '18 at 17:28










  • $begingroup$
    @StubbornAtom yes, it's right to call it a test
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29










  • $begingroup$
    @StubbornAtom added
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29


















  • $begingroup$
    You don't have to use the student's t-distribution unless your sample is small.
    $endgroup$
    – Frpzzd
    Dec 11 '18 at 17:07










  • $begingroup$
    @Frpzzd actually I'm interested in the correctness of my proof
    $endgroup$
    – openspace
    Dec 11 '18 at 17:18












  • $begingroup$
    What do you mean by "criteria for $H_0$..."? Are you deriving a test for testing $H_0$ against some $H_1$?
    $endgroup$
    – StubbornAtom
    Dec 11 '18 at 17:28










  • $begingroup$
    @StubbornAtom yes, it's right to call it a test
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29










  • $begingroup$
    @StubbornAtom added
    $endgroup$
    – openspace
    Dec 11 '18 at 17:29
















$begingroup$
You don't have to use the student's t-distribution unless your sample is small.
$endgroup$
– Frpzzd
Dec 11 '18 at 17:07




$begingroup$
You don't have to use the student's t-distribution unless your sample is small.
$endgroup$
– Frpzzd
Dec 11 '18 at 17:07












$begingroup$
@Frpzzd actually I'm interested in the correctness of my proof
$endgroup$
– openspace
Dec 11 '18 at 17:18






$begingroup$
@Frpzzd actually I'm interested in the correctness of my proof
$endgroup$
– openspace
Dec 11 '18 at 17:18














$begingroup$
What do you mean by "criteria for $H_0$..."? Are you deriving a test for testing $H_0$ against some $H_1$?
$endgroup$
– StubbornAtom
Dec 11 '18 at 17:28




$begingroup$
What do you mean by "criteria for $H_0$..."? Are you deriving a test for testing $H_0$ against some $H_1$?
$endgroup$
– StubbornAtom
Dec 11 '18 at 17:28












$begingroup$
@StubbornAtom yes, it's right to call it a test
$endgroup$
– openspace
Dec 11 '18 at 17:29




$begingroup$
@StubbornAtom yes, it's right to call it a test
$endgroup$
– openspace
Dec 11 '18 at 17:29












$begingroup$
@StubbornAtom added
$endgroup$
– openspace
Dec 11 '18 at 17:29




$begingroup$
@StubbornAtom added
$endgroup$
– openspace
Dec 11 '18 at 17:29










1 Answer
1






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oldest

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0












$begingroup$

I think you're testing the hypothesis $H_0 : mu_1 = mu_2$. At least you test statistic seems to suggest so. Mind you, I am only a beginner in this field, so you might be right and I might be wrong. Also, I know only about the equality case (usually the $H_0$ is based on equality, right-tailed, left-tailed or two-tailed). Anyways, here goes



For testing $H_{0} : sigma_{1}^{2}= sigma_{2}^{2}$, the appropriate test statistic is



$$F_0 = frac{S_1^2}{S_2^2}$$



where the reference distribution of $F_0$ is the $F$ distribution with $n-1$ degrees of freedom for numerator and $m-1$ degrees of freedom for denominator. The null hypothesis would be rejected if $F_0 gt F_{alpha/2, n-1,m-1}$ or if $F_0 lt F_{1-(alpha/2), n-1,m-1}$



You can read more about it in the book Design of Experiments by Montgomery, Chapter 2, the ending section.






share|cite|improve this answer











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    1 Answer
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    active

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    0












    $begingroup$

    I think you're testing the hypothesis $H_0 : mu_1 = mu_2$. At least you test statistic seems to suggest so. Mind you, I am only a beginner in this field, so you might be right and I might be wrong. Also, I know only about the equality case (usually the $H_0$ is based on equality, right-tailed, left-tailed or two-tailed). Anyways, here goes



    For testing $H_{0} : sigma_{1}^{2}= sigma_{2}^{2}$, the appropriate test statistic is



    $$F_0 = frac{S_1^2}{S_2^2}$$



    where the reference distribution of $F_0$ is the $F$ distribution with $n-1$ degrees of freedom for numerator and $m-1$ degrees of freedom for denominator. The null hypothesis would be rejected if $F_0 gt F_{alpha/2, n-1,m-1}$ or if $F_0 lt F_{1-(alpha/2), n-1,m-1}$



    You can read more about it in the book Design of Experiments by Montgomery, Chapter 2, the ending section.






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      I think you're testing the hypothesis $H_0 : mu_1 = mu_2$. At least you test statistic seems to suggest so. Mind you, I am only a beginner in this field, so you might be right and I might be wrong. Also, I know only about the equality case (usually the $H_0$ is based on equality, right-tailed, left-tailed or two-tailed). Anyways, here goes



      For testing $H_{0} : sigma_{1}^{2}= sigma_{2}^{2}$, the appropriate test statistic is



      $$F_0 = frac{S_1^2}{S_2^2}$$



      where the reference distribution of $F_0$ is the $F$ distribution with $n-1$ degrees of freedom for numerator and $m-1$ degrees of freedom for denominator. The null hypothesis would be rejected if $F_0 gt F_{alpha/2, n-1,m-1}$ or if $F_0 lt F_{1-(alpha/2), n-1,m-1}$



      You can read more about it in the book Design of Experiments by Montgomery, Chapter 2, the ending section.






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        I think you're testing the hypothesis $H_0 : mu_1 = mu_2$. At least you test statistic seems to suggest so. Mind you, I am only a beginner in this field, so you might be right and I might be wrong. Also, I know only about the equality case (usually the $H_0$ is based on equality, right-tailed, left-tailed or two-tailed). Anyways, here goes



        For testing $H_{0} : sigma_{1}^{2}= sigma_{2}^{2}$, the appropriate test statistic is



        $$F_0 = frac{S_1^2}{S_2^2}$$



        where the reference distribution of $F_0$ is the $F$ distribution with $n-1$ degrees of freedom for numerator and $m-1$ degrees of freedom for denominator. The null hypothesis would be rejected if $F_0 gt F_{alpha/2, n-1,m-1}$ or if $F_0 lt F_{1-(alpha/2), n-1,m-1}$



        You can read more about it in the book Design of Experiments by Montgomery, Chapter 2, the ending section.






        share|cite|improve this answer











        $endgroup$



        I think you're testing the hypothesis $H_0 : mu_1 = mu_2$. At least you test statistic seems to suggest so. Mind you, I am only a beginner in this field, so you might be right and I might be wrong. Also, I know only about the equality case (usually the $H_0$ is based on equality, right-tailed, left-tailed or two-tailed). Anyways, here goes



        For testing $H_{0} : sigma_{1}^{2}= sigma_{2}^{2}$, the appropriate test statistic is



        $$F_0 = frac{S_1^2}{S_2^2}$$



        where the reference distribution of $F_0$ is the $F$ distribution with $n-1$ degrees of freedom for numerator and $m-1$ degrees of freedom for denominator. The null hypothesis would be rejected if $F_0 gt F_{alpha/2, n-1,m-1}$ or if $F_0 lt F_{1-(alpha/2), n-1,m-1}$



        You can read more about it in the book Design of Experiments by Montgomery, Chapter 2, the ending section.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 12 '18 at 4:03

























        answered Dec 11 '18 at 17:46









        Sauhard SharmaSauhard Sharma

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