latex hypothesis numbering

Custom Theorem, Assumption, and Lemma Numbering in Latex

Examples of customized theorem names.

I saw this kind of alternative assumption definition in other papers, such as:

• Assumption 1: …
• Assumption 2: …
• Assumption 2’: …

Here we define two alternative assumptions. How to do it in Latex with `\newtheorem’ command. For example, if we define an assumption in the following way:

I now define the following assumptionp :

For alternative assumption, we define assumptionalt as a new theorem. It uses assumption as parent counter ([assumption]). Then we define a new environment assumptionp . It takes one argument, which will allow us to customize the section name. Examples:

It looks like this:

I derived all these from StackOverflow. It took me a while to understand it. You can easily replace assumption with theorem, lemma, etc.

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Latex for hypothesis testing

Jeffrey liang.

R Code can be found in here

Distribution

\[ P(\mathcal{X}=k)\ = {n \choose k}p^k(1-p)^{n-k} \]

Normal Distribution

\(P(\mathcal{X}=k)\ = \frac{1}{\sqrt{2\pi}\sigma}*e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)

One Group t-test

Left tailed.

\(H_0\) : there’s no difference betwee sample mean and the true mean

\(H_1\) : the sample mean is lower than the true mean

\(\bar{x} = \frac{\sum_{i=1}^n{x_i}}{n}\)

\(s_d = \sqrt{\sum_{i=1}^n{(x_i - \bar{x})^2}/(n-1)}\)

\(t = \frac{\bar{x} - 0}{s_d/\sqrt{n}}\)

\(Reject ~ H_0 ~ if ~ t<t_{n-1,\alpha}\)

\(Fail ~ reject ~ H_0 ~ if ~ t>t_{n-1,\alpha}\)

\(CI : (-\infty,\bar{x} - t_{df,\alpha}*sd/\sqrt{n})\)

right tailed

\(H_1\) : the sample mean is greater than the true mean

\(Reject ~ H_0 ~ if ~ t>t_{n-1,1-\alpha}\)

\(Fail ~ reject ~ H_0 ~ if ~ t<t_{n-1,1-\alpha}\)

\(CI : (\bar{x} + t_{df,1-\alpha}*sd/\sqrt{n},\infty)\)

\(H_1\) : the sample mean is different from the true mean

\(Reject ~ H_0 ~ if ~ |t|<t_{n-1,1-\alpha/2}\)

\(Fail ~ reject ~ H_0 ~ if ~ |t|<t_{n-1,1 - \alpha/2}\)

\(CI : (\bar{x} - t_{df,1-\alpha/2}*sd/\sqrt{n} ,\bar{x} + t_{df,1-\alpha/2}*sd/\sqrt{n})\)

\(H_0\) : there’s no difference betwee difference

\(H_1\) : the difference is different

\(\bar{d} = \frac{\sum_{i=1}^n{d_i}}{n}\)

\(s_d = \sqrt{\sum_{i=1}^n{(d_i - \bar{d})^2}/(n-1)}\)

\(t = \frac{\bar{d} - 0}{s_d/\sqrt{n}}\)

\(CI : (\bar{d} - t_{df,1-\alpha/2}*sd/\sqrt{n} ,\bar{d} + t_{df,1-\alpha/2}*sd/\sqrt{n})\)

Two Group testing

Variance test.

\(H_0\) : the variances between group are equal(no difference)

\(H_1\) : the variances between group are not equal(there’s difference)

\(s_{x_1} = \sqrt{\sum_{i=1}^{n_1}(x_i - \bar{x_1})^2/(n_1-1)}\)

\(s_{x_2} = \sqrt{\sum_{j=1}^{n_2}(x_i - \bar{x_2})^2/(n_2-1)}\)

\(F = s_1^2/s_2^2 \sim F_{n_1-1,n_2-1}\)

\(Reject ~ H_0 ~ if ~ F>F_{n_1-1,n_2-1,1-\alpha/2} ~ OR ~ F<F_{n_1-1,n_2-1,\alpha/2}\)

\(Fail ~ reject ~ H_0 ~ if ~ F_{n_1-1,n_2-1,\alpha/2}<F<F_{n_1-1,n_2-1,1-\alpha/2}\)

t-test with equal variance

\(H_0\) : the means between group are equal(no difference)

\(H_1\) : the means between group are not equal(there’s difference)

\(s_{pool} = \frac{(n_1-1)s_1 + (n_2 -1)s_2}{n_1+n_2-2}\)

\(t = \frac{\bar{X_1} - \bar{X_2}}{s_{pool}\times\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\)

\(Reject ~ H_0 ~ if ~ |t|>t_{df,1-\alpha/2}\)

\(Fail ~ reject ~ H_0 ~ if ~ |t|<t_{df,1-\alpha/2}\)

\(CI = (\bar{X_1}-\bar{X_2} ~ - ~ t_{df,1-\alpha/2}s_{pool}/\sqrt{1/n_1+1/n_2},\bar{X_1}-\bar{X_2} ~ + ~ t_{df,1-\alpha/2}s_{pool}/\sqrt{1/n_1+1/n_2} )\)

t-test with unequal variance

\(\bar{X_1} - \bar{X_2} \sim N(\mu_1-\mu_2,\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2})\) if we know the population variance

\(t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}~\sim t_{d''}\)

\(d' = round(d'') = \frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})^2}{\frac{s_1^2}{n_1}^2/(n_1-1)+\frac{s_2^2}{n_2}^2/(n_2-1)}\)

\(CI = (\bar{X_1}-\bar{X_2} ~ - ~ t_{df,1-\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}},\bar{X_1}-\bar{X_2} ~ + ~ t_{df,1-\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}})\)

Multigroup Camparison

\(H_0\) : there’s no difference between groups

\(H_1\) : at least one group is different from the other groups

\(Between~Sum~of~Square = \sum_{i=1}^k\sum_{j=1}^{n_i}(\bar{y_i} - \bar{\bar{y}})^2=\sum_i^kn_i\bar{y_i}^2-\frac{y_{..}^2}{n}\)

\(Within~Sum~of~Square = \sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y_i})^2=\sum_i^k(n_i-1)s_i^2\)

\(Between~Mean~Square = \frac{\sum_{i=1}^k\sum_{j=1}^{n_i}(\bar{y_i} - \bar{\bar{y}})^2}{k-1}\)

\(Within~Mean~Square = \frac{\sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y_i})^2}{n-k}\)

\(F_{statistics} = \frac{Between~Mean~Square}{Within~Mean~Square} \sim F(k-1,n-k)\)

\(Reject ~ H_0 ~ if ~ F>F_{k-1,n-k,1-\alpha}\)

\(Fail ~ reject ~ H_0 ~ if ~F<F_{k-1,n-k,1-\alpha}\)

Proportion testing

Normal approximation, homogeneity chi-sq test.

\(H_0 :p_{1j} =p_{2j}=...=p_{ij}\) the proportion among \(group_i\) are equal …

\(H_1\) : For at least one column there’re two row i and i’ where the proability are not the same.

\(\mathcal{X}^2 = \sum_i^{row}\sum_j^{col}\frac{(n_{ij}-E_{ij})^2}{E_{ij}} \sim \mathcal{X}^2_{df = (row-1)\times(col-1)}\)

\(Reject ~ H_0 ~ if ~ \mathcal{X}^2>\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}\)

\(Fail ~ reject ~ H_0 ~ if ~\mathcal{X}^2<\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}\)

Independent test

\(H_0\) : Group A and Group B are independet

\(H_1\) : Group A and Group B are dependet/associate

Fisher Exact

Mcnemar test.

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Formatting Annotations with LaTeX

The Hypothesis editor supports LaTeX formatting markup, allowing the use of advanced math and scientific typography within annotations.

Also see our articles on adding links and images , embedding videos , and using Markdown .

If you’re wondering what LaTeX is or what it’s used for, here’s a pretty good primer .

We use the KaTeX library to render LaTeX. Here is a full list of supported functions and notation .

To use LaTeX formatting in Hypothesis, simply begin and end the markup with two dollar-signs ( $$ ). For example: $$\sum_{\mathclap{1\le i\le n}} x_{i}$$

You can also use the editor toolbar:

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Customizing Theorem, Figure, And Table Numbering In Latex

Defining your own counters.

LaTeX provides several built-in counters for numbering elements like equations, figures, and tables. However, you may want to define your own counter variables to create custom numbering schemes.

How to Create New Counters

Use the \newcounter command to define a new counter, specifying the name of the counter. For example:

This creates a new counter called “theorem” that can be used to number theorem-like environments.

Table of Contents

Incrementing and Resetting Counters

The \stepcounter command increments a counter variable by 1. To increment the “theorem” counter use:

You can reset a counter to 0 or any other number with the \setcounter command:

Setting Counter Values

To manually set a counter value, pass the desired number as an argument to \setcounter. For example, to set the theorem counter to 5:

This will override the current value of the counter.

Numbering Equations

LaTeX provides built-in support for numbering equations. Here are some ways to customize equation numbering.

Using the Equation Counter

The equation counter is automatically incremented each time you enter a math environment like \[ … \] or \begin{equation}. To number an equation, use the \label and \ref commands:

Changing Equation Number Formats

To change the way equation numbers are formatted, redefine \theequation. For example:

Will format equation numbers as simple numerals like “1”, “2”, “3” instead of with parentheses.

Restarting Equation Numbers by Section

To restart equation numbering at 1 in each new section, use:

Put this in the preamble before \begin{document}.

Customizing Theorem Environments

LaTeX’s built-in theorem-like environments such as \begin{theorem} are numbered sequentially. Here’s how to customize them.

Modifying the Theorem Style

To change the way theorems are formatted, redefine the \newtheoremstyle command. For example, to produce bold theorem headings, use:

Defining New Theorem-like Environments

Use \newtheorem to create additional theorem-like environments sharing the same counter:

This defines a “proposition” environment numbered like theorems.

Setting Separate Theorem Number Sequences

To number different theorem types independently, define a new counter for each one:

Now propositions and theorems will have separate numbering.

Formatting Tables and Figures

The presentation of tables and figures can be customized as well.

Changing the Numbering Style

Redefine \thefigure and \thetable to change the figure and table numbering format:

This will number figures with Roman numerals and tables with Arabic numerals.

Setting the Number Format

Use #1, #2 etc. within \thefigure or \thetable to insert padding:

Figures will be numbered as 1.1, 1.2, etc.

Adding Prefixes to Labels

You can append prefixes like “Fig” to figure numbers:

Figures will then be labeled Fig1, Fig2, and so on.

Restarting Numbering by Section

To restart figure and table numbering in each section, use:

Troubleshooting Numbering Issues

Sometimes figure, table, equation or theorem numbers may act unpredictably. Here is how to fix some common issues.

Dealing with Missing Numbers

If cross-references show missing numbers like “Figure ??”, it usually means the figure or label was changed after the reference was written. Compile the document twice to refresh the references.

Understanding LaTeX’s Counters

Counters hold values that increase or reset automatically in LaTeX. If a custom counter behaves strangely, check if the underlying LaTeX counters were modified.

Resetting and Clearing Counters

To start a counter over from 1, \setcounter can be used:

To clear a counter value completely, use:

Additional Numbering Customizations

Here are a few other ways LaTeX allows customizing numbering behavior.

Numbering Lines in Documents

The lineno and nolineno packages can add line numbers alongside text paragraphs for referencing.

Adding Line Numbers to Equations

The equaton and leqno packages can number individual lines within multiline equations.

Custom List Numbering Formats

Enumerated lists can be customized to use letters, roman numerals, etc. with packages like enumerate.

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LaTeX forum ⇒ Math & Science ⇒ Alternative numbering Definitions 1a 1b

Alternative numbering definitions 1a 1b.

Post by bartvde » Fri Nov 26, 2010 1:50 pm

Post by frabjous » Fri Nov 26, 2010 2:16 pm

Re: Alternative numbering Definitions 1a 1b

Post by bartvde » Fri Nov 26, 2010 3:56 pm

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• Page numbering

Document page numbers can be typeset using a particular style , such as using Arabic or Roman numerals, and typeset at a particular page location —usually within headers or footers. This article shows how the style and location of page numbers can be changed:

• the style of page numbers can be changed by the \pagenumbering command;
• the location of page numbers can be changed using the fancyhdr package .
• 1.1 Example to demonstrate page number styles
• 2 Book class: using two styles of page number
• 3 Article class: using two styles of page number
• 4 Customizing page numbers with the fancyhdr package

Setting the style of page numbers

The style of page numbers can be changed using the command

where ⟨style⟩ is one of

• arabic : use Arabic numerals (1, 2, 3, ...)
• alph : use lowercase letters (a, b, c, ...)
• Alph : use uppercase letters (A, B, C, ...)
• roman : use lowercase roman numerals (i, ii, iii, ...)
• Roman : use uppercase roman numerals (I, II, III, ...)

Example to demonstrate page number styles

The following example typesets a table of contents followed by 5 pages, each of which demonstrates one of the ⟨style⟩ options for page numbers. Internally, LaTeX uses a so-called counter to record the current page number. A counter is the name of a LaTeX variable used to store an integer value—see the Overleaf Counters article for more detail and examples.

 Open this example in Overleaf

The next graphic shows the table of contents produced by this document. Note how the \pagenumbering command has reset the starting page number to the initial value for each style: i for the roman style, A for the Alph style, I for the Roman style and so forth:

Here are two sample pages produced by this example; the first image shows page numbers as uppercase Roman numbers, the second demonstrates page numbers as uppercase letters:

Book class: using two styles of page number

The first few pages of a book , which include the copyright page, title or half-title page, any Foreword or Preface and table of contents, are collectively called the preliminary pages, or prelims for short—another term for those pages is front matter . Traditionally, prelim pages are numbered using lowercase Roman numerals with the main (body) pages being numbered using Arabic numerals. The book document class contains commands to assist with this, as the following example demonstrates.

The table of contents produced by this example demonstrates the use of Roman numerals for the front matter and Arabic numerals for the main (body) matter:

The commands that control the page numbering are:

• \frontmatter : Pages after this command and before the command \mainmatter , will be numbered with lowercase Roman numerals.
• \mainmatter : This will restart the page counter and change the style to Arabic numbers.

Article class: using two styles of page number

As noted above, the current page number is stored in a LaTeX counter variable which is called page . The value stored in page , or any other counter variable, can be set to a specific value using the \setcounter command:

where the counter variable ⟨countvar⟩ is set to the value ⟨intval⟩ . For example, to set the page counter to 3 you would write

Other commands to change counter variables include \addtocounter and \stepcounter :

• \addtocounter{⟨countvar⟩}{⟨increment⟩} adds an amount ⟨increment⟩ to the counter variable ⟨countvar⟩ . Note : ⟨increment⟩ can be positive, to increase the counter value, or negative to decrease it.
• \steptocounter{⟨countvar⟩} adds 1 to the counter variable ⟨countvar⟩

This hypothetical example modifies the page counter to plan a table of contents, including sections that haven't been written and whose page count is estimated. It also uses \pagenumbering{roman} and \pagenumbering{Arabic} to set the stye of page numbers.

 Open this example in Overleaf.

This example produces the following table of contents, showing the results of altering the value of the page counter:

Customizing page numbers with the fancyhdr package

The fancyhdr package can be used to customize the location and format of page numbers; for example, placing them at specific positions within the header or footer. The Overleaf article Headers and footers explores this in detail, with many examples we won't reproduce here.

Here we'll give one example of writing the current page number in the context of the total page count, such as Page X of Y where Y is the total number of document pages obtained using the lastpage package .

The following image shows one of the pages produced by this example—note Page 3 of 6 on the right-hand side of the footer:

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