Find a test for $H_{0} : sigma_{1}^{2} ne sigma_{2}^{2}$, against $H_{1} : sigma_{1}^{2} =sigma_{2}^{2}$
$begingroup$
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
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
$endgroup$
|
show 4 more comments
$begingroup$
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
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
$endgroup$
$begingroup$
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
|
show 4 more comments
$begingroup$
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
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
$endgroup$
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
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
edited Dec 11 '18 at 17:53
openspace
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
asked Dec 11 '18 at 16:58
openspaceopenspace
3,4452822
3,4452822
$begingroup$
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
|
show 4 more comments
$begingroup$
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
|
show 4 more comments
1 Answer
1
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$begingroup$
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.
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
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.
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
$endgroup$
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.
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
edited Dec 12 '18 at 4:03
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
answered Dec 11 '18 at 17:46
Sauhard SharmaSauhard Sharma
953318
953318
add a comment |
add a comment |
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$begingroup$
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