Dear Students as we have discussed in class following tables are summary of Statistical Hypothesis tests. The PDF version of the same is available. Send your comments for any queries. Thank You.
Let $X_{1},X_{2},\ldots X_{n}$ be independent and ideally distributed random variables according to $N(\mu,\sigma^{2})$ where $\sigma^{2}$ is unknown. We wish to test a hypothesis of the type $\sigma^{2} \geq \sigma_{0}^{2}$ or,$\sigma^{2} \leq \sigma_{0}^{2}$ or $\sigma^{2} = \sigma_{0}^{2}$, where $\sigma_{0}^{2}$ is some given positive number. Let $\bar{X} = \dfrac{\sum\limits_{i=1}^{n}X_{i}}{n}$ and $S^{2} = \dfrac{\sum\limits_{i=1}^{n}X_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{n}X_{i}\right)^{2}}{n}}{n-1}$
We summarize the tests in the following table:
The test is called Chi-Square test
Let $X_{1},X_{2},\ldots X_{n}$ be independent and ideally distributed random variables according to $N(\mu,\sigma^{2})$.Let $\bar{X} = \dfrac{\sum\limits_{i=1}^{n}X_{i}}{n}$ and $S^{2} = \dfrac{\sum\limits_{i=1}^{n}X_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{n}X_{i}\right)^{2}}{n}}{n-1}$
We summarize the tests in the following table:
The test is called t-test
Let $X_{1},X_{2},\ldots X_{m}$ and $Y_{1},Y_{2},\ldots Y_{n}$ be independent random samples distributed according to $N(\mu_{1},\sigma_{1}^{2})$ and $N(\mu_{2},\sigma_{2}^{2})$ respectively.Also
$\bar{X} = \dfrac{\sum\limits_{i=1}^{m}X_{i}}{m}$ , $S_{1}^{2} = \dfrac{\sum\limits_{i=1}^{m}X_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{m}X_{i}\right)^{2}}{m}}{m-1}$
$\bar{Y} = \dfrac{\sum\limits_{i=1}^{n}Y_{i}}{n}$ $S_{2}^{2} = \dfrac{\sum\limits_{i=1}^{n}Y_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{n}Y_{i}\right)^{2}}{n}}{n-1}$
and $S_{p}^{2} = \dfrac{(m-1)S_{1}^{2}+(n-1)S_{2}^{2}}{m+n-2}$
$S_{p}^{2}$ is sometimes called pooled sample variance
The following table summarize the test:
The test is called t-test
Let $X_{1},X_{2},\ldots X_{m}$ and $Y_{1},Y_{2},\ldots Y_{n}$ be independent random samples distributed according to $N(\mu_{1},\sigma_{1}^{2})$ and $N(\mu_{2},\sigma_{2}^{2})$ respectively.The following table summarize the test:
The test is called F-test
One- Way ANOVA
Null Hypothesis: $H_{0}:\mu_{1} = \mu_{2} = \mu_{3}=$ …$=\mu_{k} = \mu$
Let G = $\sum\limits_{i=1}^{n_{j}}\sum\limits_{j=1}^{k}x_{ij}$,and $n = \sum\limits_{j=1}^{k}n_{j}$,
Define CF = $\dfrac{G^{2}}{n}$,
Therefore
$\text{TSS} = \sum\limits_{i=1}^{n_{j}}\sum\limits_{j=1}^{k}x_{ij}^{2}- CF$, $\text{BSS} = \sum\limits_{j=1}^{k}\left(\dfrac{T_{j}^{2}}{n_{j}}\right)-CF$, and WSS = TSS - BSS
Reject $H_{0}$ if Fratio > Ftable
Let $X_{1},X_{2},\ldots,X_{n}$ denotes random sample of size $n$ from Normal Population. The population has a mean $\mu$ and standard deviation $\sigma$.
Confidence Interval for population mean $\mu$ and Normal distribution
If the sample data conforms to the normal distribution, a $(1-\alpha)100 \%$-level two sided confidence interval for mean $\mu$ is given by $\left(\bar{x}-z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}},\bar{x}+z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\right)$
If $\sigma$ is not known, then
$\left(\bar{x}-t_{n-1,\alpha/2}\dfrac{S}{\sqrt{n}},\bar{x}+t_{n-1,\alpha/2}\dfrac{S}{\sqrt{n}}\right)$
is a $(1-\alpha)100 \%$-level two sided confidence interval for mean $\mu$.
Confidence Interval for population variance $\sigma^{2}$ and Normal distribution
If the sample data conforms to the normal distribution, a $(1-\alpha)100 \%$-level two sided confidence interval for variance $\sigma^{2}$ when mean $\mu$ is known is
$\left(\dfrac{\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2}}{\chi_{n,\alpha/2}^{2}},\dfrac{\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2}}{\chi_{n,1-\alpha/2}^{2}}\right)$
If $\mu$ is unknown then
$\left(\dfrac{(n-1)S^{2}}{\chi_{n-1,\alpha/2}^{2}},\dfrac{(n-1)S^{2}}{\chi_{n-1,1-\alpha/2}^{2}}\right)$
Let $X_{1},X_{2},\ldots,X_{n_{1}}$ and $Y_{1},Y_{2},\ldots,Y_{n_{2}}$ be independent random samples of from two Normal Populations. The means and standard deviations for these populations are $\mu_{1},\mu_{2},\sigma_{1},\sigma_{2}$ respectively.
If sample data confirms the Normal distribution, a $(1-\alpha)100\%$-level two sided confidence interval for $(\mu_{1}-\mu_{2})$ when both $\sigma_{1}^{2},\sigma_{2}^{2}$ are known is given by,
$\left((\bar{x}-\bar{y})-z_{\alpha/2}\sqrt{\dfrac{\sigma_{1}^{2}}{n_{1}}+\dfrac{\sigma_{2}^{2}}{n_{2}}},(\bar{x}-\bar{y})+z_{\alpha/2}\sqrt{\dfrac{\sigma_{1}^{2}}{n_{1}}+\dfrac{\sigma_{2}^{2}}{n_{2}}}\right)$
a $(1-\alpha)100\%$-level two sided confidence interval for $(\mu_{1}-\mu_{2})$ when both $\sigma_{1}^{2},\sigma_{2}^{2}$ are unknown is given by,
$\left((\bar{x}-\bar{y})-t_{n_{1}+n_{2}-2,\alpha/2}S_{p}\sqrt{\dfrac{1}{n_{1}}+\dfrac{1}{n_{2}}},(\bar{x}-\bar{y})+t_{n_{1}+n_{2}-2,\alpha/2}S_{p}\sqrt{\dfrac{1}{n_{1}}+\dfrac{1}{n_{2}}}\right)$
Where $S_{p}^{2}=\dfrac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}$
Let $X_{1},X_{2},\ldots X_{n}$ be independent and ideally distributed random variables according to $N(\mu,\sigma^{2})$ where $\sigma^{2}$ is unknown. We wish to test a hypothesis of the type $\sigma^{2} \geq \sigma_{0}^{2}$ or,$\sigma^{2} \leq \sigma_{0}^{2}$ or $\sigma^{2} = \sigma_{0}^{2}$, where $\sigma_{0}^{2}$ is some given positive number. Let $\bar{X} = \dfrac{\sum\limits_{i=1}^{n}X_{i}}{n}$ and $S^{2} = \dfrac{\sum\limits_{i=1}^{n}X_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{n}X_{i}\right)^{2}}{n}}{n-1}$
We summarize the tests in the following table:
| Reject H0 at level $\alpha$ if | |||||
| $H_{0}$ | $H_{1}$ | $\mu$ known | $\mu$ unknown | ||
| 1 | $\sigma \geq \sigma_{0} $ | $\sigma < \sigma_{0}$ | $\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2}\leq \chi_{n,1-\alpha}^{2}\sigma_{0}^{2} $ | $s^{2} \leq \dfrac{\sigma_{0}^{2}}{n-1}\chi_{n-1,1-\alpha}^{2}$ | |
| 2 | $\sigma \leq \sigma_{0} $ | $\sigma > \sigma_{0}$ | $\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2}\geq \chi_{n,\alpha}^{2}\sigma_{0}^{2} $ | $s^{2} \geq \dfrac{\sigma_{0}^{2}}{n-1}\chi_{n-1,\alpha}^{2}$ | |
| 3 | $\sigma = \sigma_{0} $ | $\sigma \neq \sigma_{0}$ | $\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2} \leq \chi_{n,1-\alpha/2}^{2}\sigma_{0}^{2}$ | $s^{2} \leq \dfrac{\sigma_{0}^{2}}{n-1}\chi_{n-1,1-\alpha/2}^{2}$ | |
| or | or | ||||
|---|---|---|---|---|---|
| $\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2}\geq \chi_{n,\alpha/2}^{2}\sigma_{0}^{2} $ | $s^{2} \geq \dfrac{\sigma_{0}^{2}}{n-1}\chi_{n-1,\alpha/2}^{2}$ | ||||
The test is called Chi-Square test
Let $X_{1},X_{2},\ldots X_{n}$ be independent and ideally distributed random variables according to $N(\mu,\sigma^{2})$.Let $\bar{X} = \dfrac{\sum\limits_{i=1}^{n}X_{i}}{n}$ and $S^{2} = \dfrac{\sum\limits_{i=1}^{n}X_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{n}X_{i}\right)^{2}}{n}}{n-1}$
We summarize the tests in the following table:
| Reject H0 at level $\alpha$ if | ||||
| $H_{0}$ | $H_{1}$ | $\sigma$ known | $\sigma$ unknown | |
| 1 | $\mu \leq \mu_{0} $ | $\mu > \mu_{0}$ | $\bar{x} \geq \mu_{0}+\dfrac{\sigma}{\sqrt{n}}z_{\alpha} $ | $\bar{x} \geq \mu_{0}+\dfrac{s}{\sqrt{n}}t_{n-1,\alpha}$ |
| 2 | $\mu \geq \mu_{0} $ | $\mu < \mu_{0}$ | $\bar{x} \leq \mu_{0}+\dfrac{\sigma}{\sqrt{n}}z_{1-\alpha} $ | $\bar{x} \leq \mu_{0}+\dfrac{s}{\sqrt{n}}t_{n-1,1-\alpha}$ |
| 3 | $\mu = \mu_{0} $ | $\mu \neq \mu_{0}$ | $\mid \bar{x}-\mu_{0} \mid \geq \dfrac{\sigma}{\sqrt{n}}z_{\alpha/2} $ | $\mid \bar{x}-\mu_{0} \mid \geq \dfrac{s}{\sqrt{n}}t_{n-1,\alpha/2}$ |
The test is called t-test
Let $X_{1},X_{2},\ldots X_{m}$ and $Y_{1},Y_{2},\ldots Y_{n}$ be independent random samples distributed according to $N(\mu_{1},\sigma_{1}^{2})$ and $N(\mu_{2},\sigma_{2}^{2})$ respectively.Also
$\bar{X} = \dfrac{\sum\limits_{i=1}^{m}X_{i}}{m}$ , $S_{1}^{2} = \dfrac{\sum\limits_{i=1}^{m}X_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{m}X_{i}\right)^{2}}{m}}{m-1}$
$\bar{Y} = \dfrac{\sum\limits_{i=1}^{n}Y_{i}}{n}$ $S_{2}^{2} = \dfrac{\sum\limits_{i=1}^{n}Y_{i}^{2}-\dfrac{\left(\sum\limits_{i=1}^{n}Y_{i}\right)^{2}}{n}}{n-1}$
and $S_{p}^{2} = \dfrac{(m-1)S_{1}^{2}+(n-1)S_{2}^{2}}{m+n-2}$
$S_{p}^{2}$ is sometimes called pooled sample variance
The following table summarize the test:
| Reject H0 at level $\alpha$ if | ||||
| $H_{0}$ | $H_{1}$ | $\sigma_{1}^{2},\sigma_{2}^{2}$ known | ||
| 1 | $\mu_{1}- \mu_{2} \leq \mu_{0} $ | $\mu_{1}- \mu_{2} > \mu_{0}$ | $\bar{x}-\bar{y} \geq \mu_{0}+z_{\alpha}\sqrt{\dfrac{\sigma_{1}^{2}}{m}+\dfrac{\sigma_{2}^{2}}{n}} $ | |
| 2 | $\mu_{1}- \mu_{2} \geq \mu_{0} $ | $\mu_{1}- \mu_{2} < \mu_{0}$ | $\bar{x}-\bar{y} \leq \mu_{0}-z_{\alpha}\sqrt{\dfrac{\sigma_{1}^{2}}{m}+\dfrac{\sigma_{2}^{2}}{n}} $ | |
| 3 | $\mu_{1}- \mu_{2} = \mu_{0}$ | $\mu_{1}- \mu_{2} \neq \mu_{0}$ | $\mid \bar{x}-\bar{y}-\mu_{0}\mid \geq z_{\alpha/2}\sqrt{\dfrac{\sigma_{1}^{2}}{m}+\dfrac{\sigma_{2}^{2}}{n}} $ | |
| Reject H0 at level $\alpha$ if | ||||
| $H_{0}$ | $H_{1}$ | $\sigma_{1}^{2},\sigma_{2}^{2}$ unknown and $\sigma_{1} =\sigma_{2}$ | ||
| 1 | $\mu_{1}- \mu_{2} \leq \mu_{0} $ | $\mu_{1}- \mu_{2} > \mu_{0}$ | $\bar{x}-\bar{y} \geq \mu_{0}+t_{m+n-2,\alpha}s_{p}\sqrt{\dfrac{1}{m}+\dfrac{1}{n}} $ | |
| 2 | $\mu_{1}- \mu_{2} \geq \mu_{0} $ | $\mu_{1}- \mu_{2} < \mu_{0}$ | $\bar{x}-\bar{y} \leq \mu_{0}-t_{m+n-2,\alpha}s_{p}\sqrt{\dfrac{1}{m}+\dfrac{1}{n}} $ | |
| 3 | $\mu_{1}- \mu_{2} = \mu_{0}$ | $\mu_{1}- \mu_{2} \neq \mu_{0}$ | $\mid \bar{x}-\bar{y} -\mu_{0}\mid \geq t_{m+n-2,\alpha/2}s_{p}\sqrt{\dfrac{1}{m}+\dfrac{1}{n}} $ | |
The test is called t-test
Let $X_{1},X_{2},\ldots X_{m}$ and $Y_{1},Y_{2},\ldots Y_{n}$ be independent random samples distributed according to $N(\mu_{1},\sigma_{1}^{2})$ and $N(\mu_{2},\sigma_{2}^{2})$ respectively.The following table summarize the test:
| Reject H0 at level $\alpha$ if | |||||
| $H_{0}$ | $H_{1}$ | $\mu_{1},\mu_{2}$ known | $\mu_{1},\mu_{2}$ unknown | ||
| 1 | $\sigma_{1}^{2} \leq \sigma_{2}^{2}$ | $\sigma_{1}^{2} > \sigma_{2}^{2}$ | $\dfrac{\sum\limits_{i=1}^{m}(x_{i}-\mu_{1})^{2}}{\sum\limits_{i=1}^{n}(y_{i}-\mu_{2})^{2}}\geq \dfrac{m}{n}F_{m,n,\alpha} $ | $\dfrac{s_{1}^{2}}{s_{2}^{2}} \geq F_{m-1,n-1,\alpha}$ | |
| 2 | $\sigma_{1}^{2} \geq \sigma_{2}^{2}$ | $\sigma_{1}^{2} < \sigma_{2}^{2} $ | $\dfrac{\sum\limits_{i=1}^{n}(y_{i}-\mu_{2})^{2}}{\sum\limits_{i=1}^{m}(x_{i}-\mu_{1})^{2}}\geq \dfrac{n}{m}F_{n,m,\alpha} $ | $\dfrac{s_{2}^{2}}{s_{1}^{2}} \geq F_{n-1,m-1,\alpha}$ | |
| 3 | $\sigma_{1}^{2} = \sigma_{2}^{2} $ | $\sigma_{1}^{2} \neq \sigma_{2}^{2} $ | $\dfrac{\sum\limits_{i=1}^{m}(x_{i}-\mu_{1})^{2}}{\sum\limits_{i=1}^{n}(y_{i}-\mu_{2})^{2}}\geq \dfrac{m}{n}F_{m,n,\alpha/2} $ | $\dfrac{s_{1}^{2}}{s_{2}^{2}} \geq F_{m-1,n-1,\alpha/2}$ | |
| or | or | ||||
|---|---|---|---|---|---|
| $\dfrac{\sum\limits_{i=1}^{m}(x_{i}-\mu_{1})^{2}}{\sum\limits_{i=1}^{n}(y_{i}-\mu_{2})^{2}}\leq \dfrac{m}{n}F_{m,n,1-\alpha} $ | $\dfrac{s_{1}^{2}}{s_{2}^{2}} \leq F_{m-1,n-1,1-\alpha/2}$ | ||||
The test is called F-test
One- Way ANOVA
Null Hypothesis: $H_{0}:\mu_{1} = \mu_{2} = \mu_{3}=$ …$=\mu_{k} = \mu$
| Group 1 | Group 2 | … | Group j | … | Group k |
| $x_{11}$ | $x_{12}$ | … | $x_{1j}$ | … | $x_{1k}$ |
| $x_{21}$ | $x_{22}$ | … | $x_{2j}$ | … | $x_{2k}$ |
| ⋮ | ⋮ | … | ⋮ | … | ⋮ |
| $x_{i1}$ | $x_{i2}$ | … | $x_{ij}$ | … | $x_{ik}$ |
| ⋮ | ⋮ | … | ⋮ | … | ⋮ |
| $x_{n_{1}1}$ | $x_{n_{2}2}$ | … | $x_{n_{j}j}$ | … | $x_{n_{k}k}$ |
| $T_{1}=\sum\limits_{i=1}^{n_{1}}x_{i1}$ | $T_{2}=\sum\limits_{i=1}^{n_{2}}x_{i2}$ | … | $T_{j}=\sum\limits_{i=1}^{n_{j}}x_{ij}$ | … | $T_{k}=\sum\limits_{i=1}^{n_{k}}x_{ik}$ |
| $T_{1}^{2}$ | $T_{2}^{2}$ | … | $T_{j}^{2}$ | … | $T_{k}^{2}$ |
| Source of Variation | SS | df | MS | Fratio | F table |
| Between Group | $\sum\limits_{j=1}^{k}\left(\dfrac{T_{j}^{2}}{n_{j}}\right)-CF$ | $k-1$ | BSS/df | MSB/MSW | $F_{\alpha,(k-1,n-k)}$ |
| Within Groups | TSS-BSS | $n-k$ | WSS/df | ||
| Total | $\sum\limits_{i=1}^{n_{j}}\sum\limits_{j=1}^{k}x_{ij}^{2}- CF$ | $n-1$ | MS | ||
Let $X_{1},X_{2},\ldots,X_{n}$ denotes random sample of size $n$ from Normal Population. The population has a mean $\mu$ and standard deviation $\sigma$.
Confidence Interval for population mean $\mu$ and Normal distribution
If the sample data conforms to the normal distribution, a $(1-\alpha)100 \%$-level two sided confidence interval for mean $\mu$ is given by $\left(\bar{x}-z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}},\bar{x}+z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\right)$
If $\sigma$ is not known, then
$\left(\bar{x}-t_{n-1,\alpha/2}\dfrac{S}{\sqrt{n}},\bar{x}+t_{n-1,\alpha/2}\dfrac{S}{\sqrt{n}}\right)$
is a $(1-\alpha)100 \%$-level two sided confidence interval for mean $\mu$.
Confidence Interval for population variance $\sigma^{2}$ and Normal distribution
If the sample data conforms to the normal distribution, a $(1-\alpha)100 \%$-level two sided confidence interval for variance $\sigma^{2}$ when mean $\mu$ is known is
$\left(\dfrac{\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2}}{\chi_{n,\alpha/2}^{2}},\dfrac{\sum\limits_{i=1}^{n}(x_{i}-\mu)^{2}}{\chi_{n,1-\alpha/2}^{2}}\right)$
If $\mu$ is unknown then
$\left(\dfrac{(n-1)S^{2}}{\chi_{n-1,\alpha/2}^{2}},\dfrac{(n-1)S^{2}}{\chi_{n-1,1-\alpha/2}^{2}}\right)$
Let $X_{1},X_{2},\ldots,X_{n_{1}}$ and $Y_{1},Y_{2},\ldots,Y_{n_{2}}$ be independent random samples of from two Normal Populations. The means and standard deviations for these populations are $\mu_{1},\mu_{2},\sigma_{1},\sigma_{2}$ respectively.
If sample data confirms the Normal distribution, a $(1-\alpha)100\%$-level two sided confidence interval for $(\mu_{1}-\mu_{2})$ when both $\sigma_{1}^{2},\sigma_{2}^{2}$ are known is given by,
$\left((\bar{x}-\bar{y})-z_{\alpha/2}\sqrt{\dfrac{\sigma_{1}^{2}}{n_{1}}+\dfrac{\sigma_{2}^{2}}{n_{2}}},(\bar{x}-\bar{y})+z_{\alpha/2}\sqrt{\dfrac{\sigma_{1}^{2}}{n_{1}}+\dfrac{\sigma_{2}^{2}}{n_{2}}}\right)$
a $(1-\alpha)100\%$-level two sided confidence interval for $(\mu_{1}-\mu_{2})$ when both $\sigma_{1}^{2},\sigma_{2}^{2}$ are unknown is given by,
$\left((\bar{x}-\bar{y})-t_{n_{1}+n_{2}-2,\alpha/2}S_{p}\sqrt{\dfrac{1}{n_{1}}+\dfrac{1}{n_{2}}},(\bar{x}-\bar{y})+t_{n_{1}+n_{2}-2,\alpha/2}S_{p}\sqrt{\dfrac{1}{n_{1}}+\dfrac{1}{n_{2}}}\right)$
Where $S_{p}^{2}=\dfrac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}$
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