91影视

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

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.

The following table shows the observed frequencies:

The following table shows the row totals and column totals:

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:

The test statistic with Yates鈥檚 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鈥檚 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 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.

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.

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