Bergman et al. conducted a study to determine, among other things, the impact that scheduling recess before or after the lunch period has on wasted food for students in grades three through five. Results were published in the online article "The Relationship of Meal and Recess Schedules to Plate Waste in Elementary Schools" (Journal of Child Nutrition and Management, Vol. 28, Issue 2). Summary statistics for the amount of food wasted, in grams, by randomly selected students are presented in the following table.

At the $1\%$significance level, do the data provide sufficient evidence to conclude that, in grades three through five, the mean amount of food wasted for lunches before recess exceeds that for lunches after recess?

Lunch before recess and lunch after recess are two populations from which sample data is available.

The significance level is$1\%$.

Population $1$: Lunch before recess, ${\overline{x}}_1=223.1,s_1=122.9$, and $n_1=889.$

Population $2$: Lunch after recess, ${\overline{x}}_2=156.6,s_2=108.1$, and $n_2=1119.$

The major goal is to come to the conclusion that the average quantity of food wasted for lunches before recess is more than the average amount of food wasted for lunch after recess.

Define null and alternate hypotheses.

Null hypotheses:$H_0:{\mu }_1\le {\mu }_2$

Alternate hypotheses:$H_a:{\mu }_1>{\mu }_2$

Hypotheses is right-tailed.

We determine the significance level as$1\%$.

Pooled standard deviation,$s_p=\sqrt{\frac{\left(n_1-1\right)s_1{}^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}}$

$\Rightarrow s_p=\sqrt{\frac{(889-1)(122.9{)}^2+(1119-1\left)\right(108.1{)}^2}{889+1119-2}}$

$\Rightarrow s_p=\sqrt{\frac{888(122.9{)}^2+1118(108.1{)}^2}{2006}}$

$\Rightarrow s_p=114.887$

Test statistic,$t_0=\frac{{\overline{x}}_1-{\overline{x}}_2}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

$\Rightarrow t_0=\frac{223.1-156.6}{114.887\sqrt{\frac{1}{889}+\frac{1}{1119}}}$

$\Rightarrow t_0=12.884$

We need to determine the critical values

Here, localid="1651301139954" role="math" $$\begin{gathered} df=n_1+n_2-2 \\ =889+1119-2 \\ =2006 \end{gathered}$$

localid="1651301154758" role="math" $\Rightarrow \mathrm{df}=2006$

Using table IV

localid="1651301144596" role="math" $$\begin{gathered} \text{Critical value,}{t}_{\alpha }=t_{0.01} \\ =2.328 \end{gathered}$$

Since $t_0>t_{\alpha }$, the test statistic falls in the right-tailed hypotheses test rejection zone. As a result, null hypotheses are rejected.