Question

The article "Growth Response in Radish to Sequential and Simultaneous Exposures of \(\mathrm{NO}_{2}\) and \(\mathrm{SO}_{2} "(\) Environmental Pollution \([1984]: 303-325\) ) compared a control group (no exposure), a sequential exposure group (plants exposed to one pollutant followed by exposure to the second four weeks later), and a simultaneous-exposure group (plants exposed to both pollutants at the same time). The article states, "Sequential exposure to the two pollutants had no effect on growth compared to the control. Simultaneous exposure to the gases significantly reduced plant growth." Let \(\bar{x}_{1}, \bar{x}_{2}\), and \(\bar{x}_{3}\) represent the sample means for the control, sequential, and simultaneous groups, respectively. Suppose that \(\bar{x}_{1}>\bar{x}_{2}>\bar{x}_{3}\). Use the given information to construct a table where the sample means are listed in increasing order, with those that are judged not to be significantly different underscored. \(15.30\) The nutritional quality of shrubs commonly used for feed by rabbits was the focus of a study summarized in the article "Estimation of Browse by Size Classes for Snowshoe Hare" (Journal of Wildlife Management [1980]: \(34-40\) ). The energy content \((\mathrm{cal} / \mathrm{g})\) of three sizes \((4 \mathrm{~mm}\) or less, \(5-7 \mathrm{~mm}\), and \(8-10 \mathrm{~mm}\) ) of serviceberries was studied. Let \(\mu_{1}, \mu_{2}\), and \(\mu_{3}\) denote the true energy content for the three size classes. Suppose that \(95 \%\) simultaneous confidence intervals for \(\mu_{1}-\mu_{2}, \mu_{1}-\mu_{3}\), and \(\mu_{2}-\mu_{3}\) are \((-10,290),(150,450)\), and \((10,310)\), respectively. How would you interpret these intervals?

Short answer

The table for the first exercise portion would have, in increasing order, the simultaneous exposure group mean, the sequential group mean (underlined, as there is no significant effect compared to the control), followed by the control group mean. For the second part of the exercise, it's inferred that the true energy content for size 1 and size 2 aren't significantly different, but both are significantly greater than the energy content of size 3.

Step-by-step solution

Organize given information of exercise part 1

Given that \(\bar{x}_{1}> \bar{x}_{2} > \bar{x}_{3}\). Considering the details of the problem, the table should look as follows:

Construct the table

In the table, the samples means are in increasing order starting from \(\bar{x}_{3}\), \(\bar{x}_{2}\), and \(\bar{x}_{3}\): \n\n Simultaneous Exposure: \(\bar{x}_{3}\) \n Sequential Exposure: \(\bar{x}_{2}\) (underscored because it's not significantly different from the control group) \n Control: \(\bar{x}_{1}\) \n\n The fact that \(\bar{x}_{2}\) is underscored indicates that it is not significantly different from the control.

Interpret intervals of exercise part 2

Given that the confidence intervals for \(\mu_{1}-\mu_{2}, \mu_{1}-\mu_{3}\), and \(\mu_{2}-\mu_{3}\) are \((-10,290),(150,450)\), and \((10,310)\), respectively. This correspond to the difference in mean energy contents between the size classes. Negative values mean the first one specified is smaller, and positive that it is larger.

Compare the intervals

If zero is not in the confidence interval for the difference between two means, then this suggests the means are significantly different. Hence: \n\n - \( \mu_1\) and \( \mu_2\) are not significantly different because both negative and positive differences are plausible (the confidence interval contains 0). \n\n - \( \mu_1\) is significantly greater than \( \mu_3\) because the confidence interval for their difference does not contain 0. \n\n - \( \mu_2\) is significantly greater than \( \mu_3\) because the confidence interval for their difference does not contain 0.