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.