Chapter 13.3, Problem 1AC

School Lunch

A nutritionist decided to see if there was a difference in the number of calories served for lunch in elementary and secondary schools. She selected a random sample of eight elementary schools and another random sample of eight secondary schools in Pennsylvania. The data are shown.

Elementary	Secondary
648	694
589	730
625	750
595	810
789	860
727	702
702	657
564	761

1. Are the samples independent or dependent?

2. What are the hypotheses?

3. What nonparametric test would you use to test the claim?

4. What critical value would you use?

5. What is the test value?

6. What is your decision?

7. What is the corresponding parametric test?

8. What assumption would you need to meet to use the parametric test?

9. If this assumption were not met, would the parametric test yield the same results?

To determine

To check: The samples are independent or dependent.

Answer to Problem 1AC

The samples are independent.

Explanation of Solution

Given info:

The data shows the number of calories served for lunch in elementary and secondary schools.

Justification:

Here, the samples are not related. Therefore, the given samples are independent.

To determine

To state: The hypotheses.

Answer to Problem 1AC

Hypotheses:

$H_0:$ There is no difference in the number of calories served for lunch in elementary and secondary schools.

Alternative hypothesis:

$H_1:$ There is a difference in the number of calories served for lunch in elementary and secondary schools.

Explanation of Solution

Here, the claim is that “there is a difference in the number of calories served for lunch in elementary and secondary schools”.

The null and alternative hypotheses are given below:

Null hypothesis:

$H_0:$ There is no difference in the number of calories served for lunch in elementary and secondary schools.

Alternative hypothesis:

$H_1:$ There is a difference in the number of calories served for lunch in elementary and secondary schools.

To determine

To decide: Whether to use a Wilcoxon signed-rank test or Wilcoxon rank sum test.

Answer to Problem 1AC

The Wilcoxon rank-sum test can be used.

Explanation of Solution

Condition:

If samples are dependent, the Wilcoxon signed-rank test can be used.

If samples are independent, the Wilcoxon rank-sum test can be used.

Here, the samples are independent. Thus, the Wilcoxon rank-sum test can be used.

To determine

To find: The critical value.

Answer to Problem 1AC

The critical value is ±1.96.

Explanation of Solution

Calculation:

Here, the test is two tailed test.

From “Appendix Table E, Cumulative standard normal distribution”, the critical value for $\alpha =0.05$ is ±1.96.

The rejection region for a two-tailed test with $\alpha =0.05$ is, $z<-z_0\left(=-1.96\right)$, $z>z_0\left(=+1.96\right)$.

To determine

To find: The test statistic value.

Answer to Problem 1AC

The test statistic is –2.1.

Explanation of Solution

Calculation:

Software procedure:

Step by step procedure to find the test statistic using the EXCEL-Megastat software:

• Choose Stat > Nonparametrics > Wilcoxon-Mann/Whitney test.
• In First Sample, select the column of Elementary in A2:A9.
• In Second Sample, select the column of Secondary in B2:B9.
• Select output ranked data, correct for ties.
• In Alternative, select not equal.
• Click OK.

Output using the EXCEL-Megastat software is given below:

Thus, the test statistic is –2.1.

To determine

To make: The decision.

Answer to Problem 1AC

The decision is “reject the null hypothesis”.

The conclusion is that there is enough evidence to support the claim that there is a difference in the number of calories served for lunch in elementary and secondary schools.

Explanation of Solution

Justification:

Decision rule:

• If $z<-z_0\left(=-1.96\right)$, then reject the null hypothesis
• If $z>z_0\left(=+1.96\right)$, then reject the null hypothesis

Here, the test statistic value is less than the critical value. That is, –2.1 < –1.96

Therefore, the null hypothesis is rejected.

Thus, there is enough evidence to support the claim that there is a difference in the number of calories served for lunch in elementary and secondary schools.

To determine

To find: The parametric equivalent test for the Wilcoxon rank sum test.

Answer to Problem 1AC

The parametric test of t for independent samples is equivalent test for the Wilcoxon rank sum test.

Explanation of Solution

Wilcoxon rank sum test: It is used for testing two samples of the independent groups come from the population having same distribution or not.

If samples are dependent, the Wilcoxon signed-rank test can be used.

If samples are independent, the Wilcoxon rank-sum test can be used.

Thus, the parametric test of t for independent samples is equivalent test for the Wilcoxon rank sum test.

To determine

To find: The assumption that need to meet to use the parametric test.

Answer to Problem 1AC

The assumption “the samples are normally distributed” is required to meet to use the parametric test.

Explanation of Solution

Non-parametric tests:

When the distribution of the population is not known or when the population distribution does not follow normality, then the non-parametric statistical tests is used to test the population parameters.

Parametric tests:

When the distribution of the population is known or when the population distribution follows normality, then the parametric statistical tests are used to test the population parameters.

Thus, the assumption “the samples are normally distributed” is required to meet to use the parametric test.

To determine

To check: If the assumption were not meet, the parametric test yields the same results.

Answer to Problem 1AC

The parametric test would yields the same results.

Explanation of Solution

The parametric test would yields the same results because t test are robust against normality.