Small-sample Proportion Test: Test Statistic and p-value

7. Hypothesis Testing: Hypothesis Test for a Population Proportion

Small-sample Proportion Test: Test Statistic and p-value

Binomial Test for a Population Proportion: Test Statistic

When the sample size is small ( $n\lt 30$ ), the Central Limit Theorem no longer applies.

In such cases, however, the number of successes $X$ can be used as the test statistic.

Under the assumption that the null hypothesis $H_0$ is true, $X$ has the binomial distribution with parameters $n$ and $\pi_0$ . That is, $X \sim B(n, \pi_0)$ .

Calculating the p-value of a Binomial Test for a Population Proportion with Statistical Software

Suppose you observe $X=k$ .

The calculation of the $p$ -value of a binomial test for $\pi$ is dependent on the direction of the test and can be performed using either Excel or R.

To calculate the $p$ -value of a binomial test for $\pi$ in Excel, make use of one of the following commands:

$\begin{array}{llll} \phantom{0}\text{Direction}&\phantom{0000}H_0&\phantom{0000}H_a&\phantom{0000000000}\text{Excel Command}\\ \hline \text{Left-tailed}&H_0:\pi \geq \pi_0&H_a:\pi \lt \pi_0&=\text{BINOM.DIST}(k, n, \pi_0,1)\\ \text{Right-tailed}&H_0:\pi \leq \pi_0&H_a:\pi \gt \pi_0&=1\text{ - }\text{BINOM.DIST}(k \text{ - }1,n,\pi_0 ,1)\\ \text{Two-tailed}&H_0:\pi = \pi_0&H_a:\pi \neq \pi_0&\text{twice the smallest of the two probabilities above}\\ \end{array}$

To calculate the $p$ -value of a binomial test for $\pi$ in R, make use of one of the following commands:

$\begin{array}{llll} \phantom{0}\text{Direction}&\phantom{0000}H_0&\phantom{0000}H_a&\phantom{000000}\text{R Command}\\ \hline \text{Left-tailed}&H_0:\pi \geq \pi_0&H_a:\pi \lt \pi_0&\text{pbinom}(k, n, \pi_0, \text{TRUE})\\ \text{Right-tailed}&H_0:\pi \leq \pi_0&H_a:\pi \gt \pi_0&\text{pbinom}(k \text{ - }1,n,\pi_0 , \text{FALSE})\\ \text{Two-tailed}&H_0:\pi = \pi_0&H_a:\pi \neq \pi_0&\text{twice the smallest of the two probabilities above}\\ \end{array}$

If $p \lt \alpha$ , reject $H_0$ and conclude $H_a$ . Otherwise, do not reject $H_0$ .

This small-sample approach can also be used for large samples, as the $p$ -values will be approximately the same for either approach.

A casino claims that their slot machines are calibrated in such a way that players have a $15\%$ chance of winning on any given spin.

A gambling commissioner is suspicious of this claim and thinks the true chances of winning are lower than $15\%$ . The commissioner plans on using a statistical test to test her suspicion.

In $22$ independent trials, the commissioner wins $2$ times.

Calculate the $p$ -value of the test and make a decision regarding $H_0: \pi \geq 0.15$ . Round your answer to $4$ decimal places. Use the significance level $\alpha = 0.05$ .

$p=0.3382$

On the basis of this $p$ -value, $H_0$ should not be rejected, because $\,p$ $\gt$ $\alpha$ .

There are a number of different ways we can calculate the $p$ -value of the test. Click on one of the panels to toggle a specific solution.

Excel Calculation

Let $X$ denote the number of observed wins out of $22$ , then $X=2$ .

A sample size of $22$ is considered large enough for the Central Limit Theorem to apply. This means that we will need to use $X$ as the test statistic.

Under the null hypothesis $H_0$ , $X$ is binomially distributed with parameters $n$ and $\pi_0$ . That is:

$X \sim B(22, 0.15)$
Assuming $X \sim B(22, 0.15)$ , the $p$ -value of a left-tailed binomial test for $\pi$ can be computed with the following Excel command:

$\begin{array}{c} = \text{BINOM.DIST}(X,n,\pi_0 ,1)\\ \downarrow\\ = \text{BINOM.DIST}(2, 22, 0.15 ,1) \end{array}$
This gives:

$p =0.3382$
Since $\,p$ $\gt$ $\alpha$ , $H_0: \pi = 0.15$ should not be rejected.

R Calculation

$X \sim B(22, 0.15)$
Assuming $X \sim B(22, 0.15)$ , the $p$ -value of a left-tailed binomial test for $\pi$ can be computed with the following R command:

$\begin{array}{c} \text{pbinom}(X,n,\pi_0 ,\text{TRUE})\\ \downarrow\\ \text{pbinom}(2, 22, 0.15 ,\text{TRUE}) \end{array}$
This gives:

$p =0.3382$
Since $\,p$ $\gt$ $\alpha$ , $H_0: \pi = 0.15$ should not be rejected.

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