Question

Using Yates’s Correction for Continuity The chi-square distribution is continuous, whereas the test statistic used in this section is discrete. Some statisticians use Yates’s correction for continuity in cells with an expected frequency of less than 10 or in all cells of a contingency table with two rows and two columns. With Yates’s correction, we replace

\(\sum \frac{{{{\left( {O - E} \right)}^2}}}{E}\)with \(\sum \frac{{{{\left( {\left| {O - E} \right| - 0.5} \right)}^2}}}{E}\)

Given the contingency table in Exercise 9 “Four Quarters the Same as $1?” find the value of the test \({\chi ^2}\)statistic using Yates’s correction in all cells. What effect does Yates’s correction have?

Short answer

The value of the \({\chi ^2}\) test statistic using the Yates’s correction factor is equal to 10.717.

There is sufficient evidence to reject the claim that whether students purchased gum or kept the money is independent of whether they were given four quarters or a $1 bill.

The effect of Yates’s correction factor is that the test statistic value decreases as compared to the test statistic value computed without using any correction factor.

Step-by-step solution

Given information

Data are given on the number of students who spent/kept the money given that they were given four quarters and a $1 bill.

Hypotheses

Null Hypothesis: The act of spending/keeping the money is independent of whether the students were given four quarters or a $1 bill.

Alternative Hypothesis: The act of spending/keeping the money is not independent of whether the students were given four quarters or a $1 bill.

Observed frequencies

The following table shows the observed frequencies:

	Purchased Gum	Kept the Money
Students given four Quarters	\({O_1} = \)27	\({O_2} = \)16
Students Given $1 Bill	\({O_3} = \)12	\({O_4} = \)34

Calculation of expected frequencies

The following table shows the row totals and column totals:

	Purchased Gum	Kept the Money	Row Total
Students given four Quarters	27	16	43
Students Given $1 Bill	12	34	46
Column total	39	50	89

The expected frequencies are computed using the given formula:

\({\rm{Expected}}\;{\rm{Frequency}} = \frac{{{\rm{Row}}\;{\rm{Total}} \times {\rm{Column}}\;{\rm{Total}}}}{{{\rm{Grand}}\;{\rm{Total}}}}\)

The following table shows the expected frequencies:

	Purchased Gum	Kept the Money
Students given four Quarters	\(\begin{aligned}{c}{E_1} = \frac{{43 \times 39}}{{89}}\\ = 18.84\end{aligned}\)	\(\begin{aligned}{c}{E_2} = \frac{{43 \times 50}}{{89}}\\ = 24.16\end{aligned}\)
Students Given $1 Bill	\(\begin{aligned}{c}{E_3} = \frac{{46 \times 39}}{{89}}\\ = 20.16\end{aligned}\)	\(\begin{aligned}{c}{E_4} = \frac{{46 \times 50}}{{89}}\\ = 25.84\end{aligned}\)

Calculation of the test statistic

The test statistic with Yates’s correction factor is computed as follows:

\(\begin{aligned}{c}{\chi ^2} = \sum \frac{{{{\left( {\left| {O - E} \right| - 0.5} \right)}^2}}}{E}\;\;\;\;\chi _{\left( {r - 1} \right)\left( {c - 1} \right)}^2\\ = \frac{{{{\left( {\left| {27 - 18.84} \right| - 0.5} \right)}^2}}}{{18.84}} + \frac{{{{\left( {\left| {16 - 24.16} \right| - 0.5} \right)}^2}}}{{24.16}} + \frac{{{{\left( {\left| {12 - 20.16} \right| - 0.5} \right)}^2}}}{{20.16}} + \frac{{{{\left( {\left| {34 - 25.84} \right| - 0.5} \right)}^2}}}{{25.84}}\\ = 10.717\end{aligned}\)

Therefore, the value of test statistic is 10.717.

The test statistic with Yates’s correction factor is computed as follows:

\(\begin{aligned}{c}{\chi ^2} = \sum \frac{{{{\left( {\left| {O - E} \right| - 0.5} \right)}^2}}}{E}\;\;\;\;\chi _{\left( {r - 1} \right)\left( {c - 1} \right)}^2\\ = \frac{{{{\left( {\left| {27 - 18.84} \right| - 0.5} \right)}^2}}}{{18.84}} + \frac{{{{\left( {\left| {16 - 24.16} \right| - 0.5} \right)}^2}}}{{24.16}} + \frac{{{{\left( {\left| {12 - 20.16} \right| - 0.5} \right)}^2}}}{{20.16}} + \frac{{{{\left( {\left| {34 - 25.84} \right| - 0.5} \right)}^2}}}{{25.84}}\\ = 10.717\end{aligned}\)

Therefore, the value of test statistic is 10.717.

Calculation of the critical value

The level of significance is 0.05.

The degrees of freedom is computed below:

\(\begin{aligned}{c}df = \left( {r - 1} \right)\left( {c - 1} \right)\\ = \left( {2 - 1} \right)\left( {2 - 1} \right)\\ = 1\end{aligned}\)

Now look at the \({\chi ^2}\) distribution table for 1 degrees of freedom with 0.05 significance level.

The critical value is 3.842.

Conclusion of the test

The calculated value of the test statistic is greater than the critical value. Therefore, the null hypothesis is rejected.

Thus, there is sufficient evidence to reject the claim that whether students purchased gum or kept the money is independent of whether they were given four quarters or a $1 bill.

It seems that there is a denomination effect.

Effect of Yates’s correction

The effect of Yates’s correction factor is that the tests statisticdecreases as compared to the test statistic computed without using any correction factor.