**There’s a $3\%$ chance that the die is fair.****There’s a $97\%$ chance that the die is fair.****There’s a $3\%$ chance that a loaded die could randomly produce the results we observed, so it’s reasonable to conclude that the die is fair.****There’s a $3\%$ chance that a fair die could randomly produce the results we observed, so it’s reasonable to conclude that the die is loaded.**

The purpose of this question is to choose the correct statement out of the given four statements about the fair die.

In statistics, testing a hypothesis is the process by which an analyst tests an assertion about a population parameter. The purpose of analysis and the type of information determines the technique used by analysts. Using statistics to investigate the world’s ideas, hypothesis testing is a systematic process.

The assertion that the event will not happen is known as the Null Hypothesis. Unless and until it is rejected, a null hypothesis does not influence the result of the survey. Logically, it is contrary to the alternate hypothesis and is denoted by $H_0$. When the null hypothesis is rejected, this implies that the alternative hypothesis is accepted. It is represented by $H_1$. The process of testing the hypothesis includes examining the sample data to check the rejection of $H_0$.

## Expert Answer

The loaded die seller claims that the outcome will be $6$.

In this question, the claim is the null or alternative hypothesis. The null hypothesis regards the fact that population proportion equals the claim value. On contrary, the alternative hypothesis regards the inverse of the null hypothesis.

The claim was tested using the hypothesis test:

$H_0: p=\dfrac{1}{6}$ and $H_1: p>\dfrac{1}{6}$

which indicates a one-tailed test.

Also, given $p-$value $=0.03$.

$p<0.03$ will result in the rejection of null hypothesis and the die will be fair if $p>0.03$.

In the given scenario, $p=0.03$ means that if a die is not loaded or fair, there is a $3\%$ chance that the sample proportion will be greater than $6$.

Hence, the statement, “There’s a $97\%$ chance that the die is fair” is correct.

## Example

An instructor figures out that $85\%$ of his pupils would like to go on the trip. He carries out a hypothesis test to see whether the percentage is the same as $85\%$. The instructor polls $50$ students and $39$ say they’d like to go on the trip. Use $1\%$ significance level to test the hypothesis to figure out the type of the test, the $p-$value, and state the conclusion.

### Solution

Formulating the hypothesis as:

$H_0:p=0.85$ and $H_1:p\neq 0.85$

The $p-$value for the two-tailed test comes out to be:

$p=0.7554$

Also, given that $\alpha=1\%=0.01$

Since $p$ is greater than $\alpha$, therefore we can conclude that there is insufficient reason to show that the proportion of pupils who want to go on a trip is less than $85\%$.