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






active

oldest

votes


















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











$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3035508%2ffind-a-test-for-h-0-sigma-12-ne-sigma-22-against-h-1-s%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    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

        953318




        953318






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3035508%2ffind-a-test-for-h-0-sigma-12-ne-sigma-22-against-h-1-s%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Bundesstraße 106

            Image
            Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück ...
            Read more

            Verónica Boquete

            Image
            Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl ...
            Read more

            Ida-Boy-Ed-Garten

            Image
            Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich ...
            Read more