**The two-way frequency table for data is summarized is given below.**

**When testing whether opinions about the use of instant replay depend on the respondent’s category, a chi-square test statistic of 27.99 and a p-value of less than 0.001 were calculated. Which of the following explanations is correct?****a) For the test, the number of degrees of freedom is 8-1=7.****b) The chi-square test should not have been used because two of the counts in the table are less than 5.****c) The null hypothesis indicates that there is a link between categories and opinions regarding the use of instant replay, and a small p-value indicates that the null hypothesis should be rejected.****d) A small p-value indicates that there is evidence of a link between the category and opinion regarding the use of instant replay.****e) The chi-square test shows that fans prefer to use instant replay.**

The question **aims** to find an **instant reply** for making difficult goal-line decisions in a **controversial soccer sport**. Advanced placement statistics includes tools and concepts such as collecting, analyzing, and drawing conclusions from data.

## Expert Answer

Chi-square test statistics are obtained if the p-value was less than $0.001$.

You must choose the correct statement from the statements provided regarding this issue.

The** correct statement** is $(d)$. In other words, a small p-value indicates evidence that there is a link between the category and opinions about the use of instant replay.

**Reasons for the wrong statements:**

**(a) **For the **test**, the **number of degrees of freedom** is obtained as

\[df=(number\:of\:rows-1)\times(number\:of\:columns-1)=(4-1)\times(2-1)=3\]

Thus, statement $a$ is **wrong**.

**(b)** Because two of the counts in the **table are less** **than** $5$, the** chi-square** test should not have been used. This statement is also** incorrect** because the assumption of the chi-square test is that the expected count for each cell is greater than $5$ rather than the observed count for each cell.

**(c) **Because of the **false null hypothesis**, this **statement is wrong.** The correct null hypothesis shows that there is no relevance between categories and opinions regarding the use of instant replay.

**(e) **The chi-square test shows that fans prefer to use instant replay.

Again, this **statement** is **incorrect** because the chi-square test in this question only shows if there is a **correlation** between the category and the opinion about the use of instant replay: fans prefer to use instant replay.

## Numerical Results

Statement $(a)$ is **wrong**.

Statement $(b)$ is **wrong**.

Statement $(c)$ is **wrong**.

Statement $(d)$ is **correct.**

Statement $(e)$ is **wrong**.

## Example

**A controversial topic in professional soccer is the use of instant replay to make difficult goal-line decisions. We asked $102$ players, fans, coaches, and representative samples of executives to comment on using instant replay to determine goal lines.**

-The **two-way frequency** table for data is **summarized** is given below.

-When testing whether opinions about the use of instant replay depend on the respondent’s category, a chi-square** test statistic** of $27.99$ and a **p-value** of less than $0.001$ were calculated. Determine if the statement is correct or not.

– For the test, the **number** of **degrees of freedom** is $7-1=6$.

The statement is **incorrect****.**