Binomial test

A binomial test is a statistical test in which the test statistic is binomially distributed . It is used to test hypotheses about characteristics that can take exactly two forms ( dichotomous characteristics).

Hypotheses and test statistics

The binomial test can be used to test the following pairs of hypotheses for the unknown probability of a characteristic in the population: ${\displaystyle p}$

test	${\displaystyle H_{0}}$	${\displaystyle H_{1}}$
two-sided	${\displaystyle p=p_{0}\,}$	${\displaystyle p\neq p_{0}}$
right side	${\displaystyle p=p_{0}}$ or ${\displaystyle p\leq p_{0}}$	${\displaystyle p>p_{0}}$
left side	${\displaystyle p=p_{0}}$ or ${\displaystyle p\geq p_{0}}$	${\displaystyle p<p_{0}}$

The test statistic shows the number of times the feature appeared in a random sample of the size . Under the null hypothesis , the test statistic is -distributed, that is ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle H_{0}\colon p=p_{0}}$ ${\displaystyle B(p_{0},n)}$

${\displaystyle P(X=i)=B(i|p_{0},n)={\binom {n}{i}}p_{0}^{i}(1-p_{0})^{n-i}}$.

Significance level and critical values

Test statistics for the binomial test, the red columns belong to the critical area.

Since the test statistics are distributed discretely, the specified level of significance can usually not be maintained. It is therefore required to choose the critical values ​​in such a way that the highest possible exact significance level applies . ${\displaystyle \alpha }$ ${\displaystyle \alpha _{\text{ex}}}$ ${\displaystyle \alpha _{\text{ex}}\leq \alpha }$

For the two-sided test, therefore, the largest and the smallest are determined as critical values for which applies ${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$

• ${\displaystyle \sum _{i=0}^{c_{1}}B(i|p_{0},n)\leq \alpha /2}$ and
• ${\displaystyle \sum _{i=c_{2}}^{n}B(i|p_{0},n)\leq \alpha /2}$.

The exact level of significance results as . The same procedure is used for the two one-sided tests . ${\displaystyle \alpha _{\text{ex}}=\sum _{i=0}^{c_{1}}B(i|p_{0},n)+\sum _{i=c_{2}}^{n}B(i|p_{0},n)}$

test	Critical values	Critical area	Limit (s)
two-sided	${\displaystyle c_{1}+1}$ and ${\displaystyle c_{2}-1}$	${\displaystyle \{0,\dotsc ,c_{1}\}\cup \{c_{2},\dotsc ,n\}}$
right side	${\displaystyle c-1}$	${\displaystyle \{c,\dotsc ,n\}}$	c = smallest value for which ${\displaystyle \sum _{i=c}^{n}B(i|p_{0},n)=\alpha _{\text{ex}}\leq \alpha }$
left side	${\displaystyle c+1}$	${\displaystyle \{0,\dotsc ,c\}}$	c = largest value for which ${\displaystyle \sum _{i=0}^{c}B(i|p_{0},n)=\alpha _{\text{ex}}\leq \alpha }$

Approximation of the distribution of the test statistic

Approximation of a binomial distribution with a normal distribution.

The binomially distributed test statistic can be approximated with another distribution. The approximation conditions necessary for this can vary depending on the literature source.

distribution	parameter	Approximation conditions
Poisson distribution ${\displaystyle X\approx Po(\lambda )}$	${\displaystyle \lambda =np_{0}}$	${\displaystyle n>10}$ and ${\displaystyle p_{0}<0{,}05}$
Normal distribution ${\displaystyle X\approx N(\mu ,\sigma ^{2})}$	${\displaystyle \mu =np_{0}}$ and ${\displaystyle \sigma ^{2}=np_{0}(1-p_{0})}$	${\displaystyle np_{0}(1-p_{0})>9}$

In the case of the approximation of the normal distribution, the test statistics can also be considered instead of the test statistics . ${\displaystyle X}$ ${\displaystyle \Pi =X/n\approx N\left(p_{0},{\tfrac {p_{0}(1-p_{o})}{n}}\right)}$

Examples

1. Clairvoyant ability versus guessing the color of a randomly chosen playing card (from a statistical test ): If repeated, a test person scores hits (correctly named color). From what number of hits should the null hypothesis be rejected and the alternative hypothesis (i.e. actual clairvoyant ability) considered more plausible ? If is correct then it is binomially distributed with parameters and 1/4. The probability of getting or more hits by guessing is then . At a significance level of 1%, the null hypothesis is rejected, if . Here is the smallest value that is for . For example for results . The test person would have to be correct in at least 36 out of 100 attempts under the conditions mentioned, so that their clairvoyant abilities are considered plausible. ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle H_{0}\colon p={\tfrac {1}{4}}}$ ${\displaystyle H_{1}\colon p>{\tfrac {1}{4}}}$ ${\displaystyle H_{0}}$ ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle \sum _{i=k}^{n}B(i|{\tfrac {1}{4}},n)}$ ${\displaystyle X\geq c}$ ${\displaystyle c}$ ${\displaystyle \sum _{i=c}^{n}B(i|{\tfrac {1}{4}},n)\leq 1\,\%}$ ${\displaystyle n=100}$ ${\displaystyle c=36}$
2. In a multiple-choice exam there are 50 questions and 4 possible answers each, of which exactly one is correct. This leads to the same question as the playing card example. The null hypothesis is that a candidate randomly ticks the answer ( ), and is the alternative hypothesis . However, this modeling assumes that there is no way to exclude certain answers as implausible. ${\displaystyle H_{0}\colon p=1/4}$ ${\displaystyle H_{1}\colon p>1/4}$
3. An urn contains 10 balls, each of which can be white or black. One would like to test the null hypothesis that all balls are white (i.e. ), and pull balls with replacement. The alternative hypothesis is and the null hypothesis is rejected as soon as one or more black balls have been drawn: The rejection area is . The type 1 error is equal to 0, since no black ball can be drawn under the null hypothesis. The rejection area is apparently independent of the significance level. The error of the 2nd kind is maximal if there is exactly one black ball, and is then . ${\displaystyle H_{0}\colon p=0}$ ${\displaystyle n}$ ${\displaystyle H_{1}\colon p>0}$ ${\displaystyle \{1,\dotsc ,n\}}$ ${\displaystyle 0{,}9^{n}}$
4. (Counterexample) Same situation, but pulling without replacing (maximum balls are pulled). As in the previous case, the type 1 error disappears. The type 2 error is determined from a hypergeometric distribution . It is a maximum for a black ball and is then . So it is not a binomial test. ${\displaystyle n=10}$ ${\displaystyle (10-n)/n}$
5. The triangle test is used to find out whether there is a difference in taste between two products and . For this purpose, three samples are arranged in an equilateral triangle, with one corner of the imaginary triangle pointing upwards. Two of the three samples belong to the product and one sample belongs to the product or vice versa. The test person's task is to find the product that only occurs once. The probability of giving the correct answer by mere guessing is . Overall, different subjects take part in the experiment. The statistical calculations are the same as in the first example, with the difference that the parameter to be tested is instead . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle n}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle {\frac {1}{4}}}$

Remarks

1. ↑ We consider the parameter range [1 / 4.1] for p in order to achieve that the null hypothesis and alternative hypothesis cover the entire parameter range. Deliberately naming the wrong color could lead to clairvoyant abilities, but we assume that the test person wants the highest possible number of hits.
2. ↑ As in the playing card example, we assume that the parameter range is [1 / 4.1] (the test object wants to achieve the highest possible number of hits).

literature

• Norbert Henze : Stochastics for beginners. 8th edition. Vieweg, 2010.
• Ulrich Krengel: Introduction to probability theory and statistics. 8th edition. Vieweg, 2005.
• Horst Rinne: Pocket book of statistics. 3. Edition. Harri Deutsch, 2003.

Web links

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