Recess and Wasted Food, E. 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?

Significance level is,$0.01$.

Null hypothesis:

$H_0:{\mu }_1={\mu }_2$

That is, there is no evidence that the mean amount of food wasted for lunches before recess exceeds that for lunches after recess.

Alternative hypothesis:

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

That is, there is evidence that the mean amount of food wasted for lunches before recess exceeds that for lunches after recess.

• In EXCEL, Select Add-Ins > PHStat > Two-Sample Tests (Summarized Data).
• Choose Pooled-Variance $t$Test.
• In Data enter $0$Under Hypothesized Difference.
• Enter $0.01$under Level of Significance.
• In population $1$Sample, enter Sample size as $889$, Sample mean as $223.1$and Sample standard deviation as $122.9$.
• In population $2$Sample, enter Sample size as $1,119$, Sample mean as $156.6$and Sample standard deviation as $108.1$.
• Choose Upper-Tail Test Under Test Options.
• In Output Options enter Pooled-Variance $t$Test for the difference between two means under title.

From the output, the value of test statistic is,$12.8835$, the critical value is$2.3282$and the$P$-value is$0$.

From the Excel add-in output, the critical value is, $2.3282$.

$P$-value:

From the Excel add-in output, the $P$-value is $0$.

Critical value approach:

Here, the value of test statistic falls in the rejection region. That is, $t\left(=12.8835\right)>t_{crit}(2.3298)$

Therefore, the null hypothesis is rejected at $1\%$level.

Thus, it can be conclude that the test results are statistically significant at$1\%$level of significance.

If $P\le \alpha$, then reject the null hypothesis. Here, the $P$-value is $0$which is lesser than the level of significance. That is, $P\left(=0\right)<\alpha \left(=0.01\right)$. Therefore, the null hypothesis is rejected at $1\%$level. Thus, it can be conclude that the test results are statistically significant at $1\%$level of significance.

Therefore, The data provide sufficient evidence to conclude that the mean amount of food wasted for lunches before recess exceeds that for lunches after recess at$1\%$level.