Question

Ecosystem Response. In the on-line paper "Changes in Lake Ice: Ecosystem Response to Global Change" (Teaching Issues and Experiments in Ecology, tiee.ecoed.net, Vol. 3), R. Bohanan et al. questioned whether there is evidence for global warming in long-term data on changes in dates of ice cover in three Wisconsin Lakes. The following table gives data, for a sample of eight years, on the number of days that ice stayed on two lakes in Madison, Wisconsin-LakMendota and Lake Monona.

a. Obtain a normal probability plot and boxplot of the paired differences.

b. Based on your results from part (a), is performing a paired t-test and give explaination

c. At the 10 % significance level, do the data provide sufficient evidence to conclude that a difference exists in the mean length of time that ice stays on these two lakes?

Short answer

Part a) The paired difference of the data is symmetric and approximately normal.

Part b) It is reasonable to use the paired t-test for comparing the two populations mean.

Part c) A difference exists in the mean length of time that ice stays on these two lakes.

Step-by-step solution

Step 1:Given information

Step 2:Explaination Part a)

Using MINITAB,we construct a normal probability plot of the paired differences as follows:

Using MINITAB, we construct a box plot of the paired differences as follows:

From the above normal probability plot and box plot, we can say that the paired difference of the data is symmetric and approximately normal.

Step 2:Explaination Part b)

As the given data is a simple random paired sample and the differences are approximately normal, therefore, it is reasonable to use the paired t-test for comparing the two populations mean.

Step 2:Explaination c)

Here we set up the null and alternative hypotheses as follows:

$H_0:{\mu }_1={\mu }_2$(There is no evidence to claim that a difference exists in the mean length of time that ice stays on these two lakes.)

Versus

$H_A:{\mu }_1\ne {\mu }_2$(There is enough evidence to claim that a difference exists in the mean length of time that ice stays on these two lakes.)

This is a two-tailed test.

Also we set up $\alpha =0.10$level of significance.

Using MINITAB, we conduct the paired t-test under the above stated null hypothesis as follows:

1 Store the data in columns named Mendota and Monona.

2 Choose Stat ? Basic Statistics ?Paired t...

3 Select the Samples in columns option button

4 Click in the First sample text box and specify Mendota

5 Click in the Second sample text box and specify Monona

6 Click the Options ... button

7 Click in theConfidence level text box and type $90$

8 Click in the Test mean text box and type$0$

9 Click thearrow button at the right of the Alternative drop-down list box and select not equal

10 Click OK twice

The output for the above test is obtained as follows:

From the above output, the test statistics is obtained as $t=0.55$and the corresponding P-value is obtained as P-value $=0.600$

As the $P$-value is less than $\alpha =0.10$level of significance, therefore, we reject the null hypothesis at $10\%$level of significance. Hence there is enough evidence to conclude that, a difference exists in the mean length of time that ice stays on these two lakes.