Question

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 (cal/g) of three sizes (4 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 first interval indicates no significant difference in energy content between size classes 4mm or less and 5-7mm. However, the other two intervals suggest significant differences in energy content: between size class 4mm or less and size class 8-10mm, and also between size class 5-7mm and size class 8-10mm.

Step-by-step solution

Interpretation of the first interval

The first interval \((-10,290)\) is for \(\mu_{1}-\mu_{2}\). This means that we are 95% confident that the true difference in energy content between size class 4mm or less (which is represented by \(\mu_{1}\)) and size class 5-7mm (which is represented by \(\mu_{2}\)) lies within -10 and 290 cal/g. Since this interval contains zero, it implies that there is no significant difference in energy content between these two size classes.

Interpretation of the second interval

The second interval \((150,450)\) is for \(\mu_{1}-\mu_{3}\), comparing size class 4mm or less and size class 8-10mm. The confidence interval suggests that the true difference in energy content between these two size classes lies in the range 150 to 450 cal/g, with 95% confidence. Because this interval does not contain zero, it suggests that there is a significant difference in energy content between these two size classes.

Interpretation of the third interval

The third interval \((10,310)\) is for \(\mu_{2}-\mu_{3}\), comparing size class 5-7mm and size class 8-10mm. We can state with 95% confidence that the true difference in energy content between these two size classes lies within 10 and 310 cal/g. Since this interval also does not contain zero, it implies that there is a significant difference in energy content between these two size classes.

Key concepts

Statistical Inference

Statistical inference is a fundamental aspect of statistics that allows researchers to draw conclusions about populations based on sample data. It hinges on the creation and interpretation of confidence intervals and hypothesis testing.

For example, in the study of serviceberries' energy content, researchers sample different size classes and compute intervals to infer the true energy content for these populations. Constructed with a specific confidence level (in this case, 95%), these confidence intervals give us a range of values that, with a certain degree of certainty, include the true population parameter.

The essence of statistical inference lies in its ability to provide a measure of uncertainty in these estimations, helping researchers understand the likelihood that their findings reflect the true nature of the data they're examining.

Energy Content Analysis

In biological studies, energy content analysis is critical for understanding the caloric value and nutritive quality of various foods, like the serviceberries analyzed in the given study.

Assessing the energy content in calories per gram (cal/g) provides insights into the diet and feeding behavior of species, such as rabbits in this case. When evaluating energy content, it's important to consider how factors like size and age of the feed source might affect its nutritional value.

The simultaneous confidence intervals calculated in the study offer a statistical comparison among different size classes of berries, suggesting which might provide higher or equal caloric values. This analysis not only influences our understanding of hare dietary preferences but also has broader ecological implications.

Significance Testing

When intervals like those found in the serviceberry study do not include zero, significance testing is used to determine whether these differences are statistically meaningful. In this context, 'significant' doesn't refer to practical importance but indicates that it's unlikely the observed difference is due to random chance.

The confidence intervals for \(\mu_{1}-\mu_{3}\) and \(\mu_{2}-\mu_{3}\), not containing zero, suggest significant differences in energy content between these size classes with a high degree of confidence. This implies that, statistically, we can expect these differences to appear consistently across repeated samples from the population. Understanding significance testing allows scientists and statisticians to make data-driven decisions and hypotheses about their research findings.