Question

It is well established that exercise is beneficial for our bodies. Recent studies appear to indicate that exercise can also do wonders for our brains, or, at least, the brains of mice. In a randomized experiment, one group of mice was given access to a running wheel while a second group of mice was kept sedentary. According to an article describing the study, "The brains of mice and rats that were allowed to run on wheels pulsed with vigorous, newly born neurons, and those animals then breezed through mazes and other tests of rodent IQ"9 compared to the sedentary mice. Studies are examining the reasons for these beneficial effects of exercise on rodent (and perhaps human) intelligence. High levels of BMP (bonemorphogenetic protein) in the brain seem to make stem cells less active, which makes the brain slower and less nimble. Exercise seems to reduce the level of BMP in the brain. Additionally, exercise increases a brain protein called noggin, which improves the brain's ability. Indeed, large doses of noggin turned mice into "little mouse geniuses," according to Dr. Kessler, one of the lead authors of the study. While research is ongoing in determining how strong the effects are, all evidence points to the fact that exercise is good for the brain. Several tests involving these studies are described. In each case, define the relevant parameters and state the null and alternative hypotheses. (a) Testing to see if there is evidence that mice allowed to exercise have lower levels of BMP in the brain on average than sedentary mice. (b) Testing to see if there is evidence that mice allowed to exercise have higher levels of noggin in the brain on average than sedentary mice. (c) Testing to see if there is evidence of a negative correlation between the level of BMP and the level of noggin in the brains of mice.

Short answer

Part (a): Null hypothesis: The mean BMP level in the brains of control mice and exercise mice is the same. Alternative hypothesis: Mice allowed to exercise have lower mean BMP levels than control mice. \n Part (b): Null hypothesis: The mean noggin level in the brains of control mice and exercise mice is the same. Alternative hypothesis: Mice allowed to exercise have higher mean noggin levels than control mice. \n Part (c): Null hypothesis: There is no correlation between the BMP and noggin levels in mouse brains. Alternative hypothesis: There is a negative correlation between the BMP and noggin levels in mouse brains.

Step-by-step solution

Part (a) - Formulate Hypothesis

In this part, the parameter of interest is the mean level of BMP in mice brains. \n Null Hypothesis (\(H_0\)): The mean level of BMP in the brains of control mice (sedentary) and the exercise group are the same. \n Alternative Hypothesis (\(H_a\)): The mean level of BMP in the brains of mice allowed to exercise is less than the control group.

Part (b) - Formulate Hypothesis

In this part, the parameter of interest is the mean level of noggin in the mice brains. \n Null Hypothesis (\(H_0\)): The mean level of noggin in the brains of control mice and the exercise group are the same. \n Alternative Hypothesis (\(H_a\)): The mean level of noggin in the brains of mice allowed to exercise is greater than the control group.

Part (c) - Formulate Hypothesis

In this part, we want to establish the relationship (correlation) between the levels of BMP and noggin in mice brains. \n Null Hypothesis (\(H_0\)): There is no correlation between the levels of BMP and noggin in the mouse brain (the correlation coefficient is zero). \n Alternative Hypothesis (\(H_a\)): There is a negative correlation between the levels of BMP and noggin in the mouse brain (the correlation coefficient is less than zero).

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial when designing scientific studies, especially in statistics. The null hypothesis, denoted as \( H_0 \), represents a statement of no effect or no difference. It's the baseline assumption for the statistical test and often suggests that any observed differences are purely due to chance rather than a systematic effect.

On the contrary, the alternative hypothesis, \( H_a \), posits that there is an effect or a difference. It is what researchers aim to support through their scientific experimentation. In the mouse exercise study, the null hypothesis suggests no mean level differences in BMP or noggin between exercise and sedentary mice, while the alternative hypotheses suggest a decrease in BMP and an increase in noggin among the exercise mice, respectively.

Mean Comparisons

When we speak of mean comparisons in statistics, we refer to evaluating whether the averages of two or more groups differ significantly. For instance, comparing the mean levels of BMP and noggin between sedentary and exercised mice involves statistical tests that can distinguish whether any observed differences in means are statistically significant or likely due to random variation.

In scientific studies like the one examined, researchers use mean comparisons to gauge the effectiveness of interventions such as exercise on brain function, and through precise hypothesis testing, we can obtain a clearer understanding of the causal relationships at play.

Correlation Coefficient Analysis

Correlation coefficient analysis is a statistical method used to measure the strength and direction of the relationship between two variables. A correlation coefficient close to +1 indicates a strong positive relationship, whereas a coefficient close to -1 indicates a strong negative relationship.

In the context of the mouse exercise study, researchers are interested in finding out if there is a significant negative correlation between BMP and noggin levels, meaning that as one variable increases, the other decreases. This statistical analysis helps to illuminate the interrelated effects of different biological processes, providing insights that extend beyond simple cause-and-effect.

Exercise Effects on Brain Function

The hypothesis that exercise can improve brain function has gained momentum through numerous studies. In particular, research on mice illustrates how physical activity can lead to profound neurological benefits, such as enhanced neurogenesis and improved cognitive abilities.

The examination of exercise-induced changes in levels of biomarkers like BMP and noggin provides concrete evidence of how physical activity can modulate brain chemistry, supporting the general hypothesis that exercise has a positive impact on brain health and function.

Scientific Experimentation in Statistics

Scientific experimentation within the discipline of statistics involves carefully designed studies that collect and analyze quantitative data. The aim is to draw meaningful conclusions about the relationships between variables within the context of a controlled environment.

In statistical experiments, such as the study on mice and exercise, every step is meticulously planned—from formulating hypotheses to deciding on appropriate statistical tests—to minimize errors and ensure reliable and valid results. By adhering to rigorous scientific methods, statisticians can provide robust evidence in support of or against a particular hypothesis.