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What is the effect of concussions on the brain? Researchers measured the brain sizes (hippocampal volume in microliters) of 25 collegiate football players with a history of clinician-diagnosed concussion and 25 collegiate football players without a history of concussion. Here are the summary statistics: 18 $$ \begin{array}{lccc} \hline \text { Group } & \text { Group Size } & \text { Mean } & \text { Standard Deviation } \\ \hline \text { Concussion } & 25 & 5784 & 609.3 \\ \hline \text { Nonconcussion } & 25 & 6489 & 815.4 \\ \hline \end{array} $$ (a) Is there evidence of a difference in mean brain size between football players with a history of concussion and those without concussions? (b) The researchers in this study stated that participants were "consecutive cases of healthy National Collegiate Athletic Association Football Bowl Subdivision Division I football athletes with \((n=25)\) or without \((n=25)\) a history of clinician-diagnosed concussion ... between June 2011 and August 2013" at a U.S. psychiatric research institute specializing in neuroimaging among collegiate football players. What effect does this information have on your conclusions in part (a)?

Step by step solution

01

Understanding the Question

We need to determine if there is a statistical difference in mean hippocampal volume between players with and without a history of concussion. This involves conducting a hypothesis test.

02

Setting Up Hypotheses

Formulate the null hypothesis (H_0) that there is no difference in means, i.e., \( \mu_1 = \mu_2 \), and the alternative hypothesis (H_a) that there is a difference, i.e., \( \mu_1 eq \mu_2 \), where \( \mu_1 \) and \( \mu_2 \) are the means for the concussion and non-concussion groups respectively.

03

Choosing the Test and Calculating the Test Statistic

Use a two-sample t-test because we compare means from two independent samples. The test statistic (t) is calculated using the formula:\[t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]where \( \bar{x}_1 = 5784 \), \( \bar{x}_2 = 6489 \), \( s_1 = 609.3 \), \( s_2 = 815.4 \), \( n_1 = 25 \), and \( n_2 = 25 \).

04

Calculating Test Statistic

Substitute the values into the formula:\[t = \frac{(5784 - 6489)}{\sqrt{\frac{609.3^2}{25} + \frac{815.4^2}{25}}}\]Calculate (t) using these values to obtain the result.

05

Interpreting the Result

Compare the calculated (t) value to the critical value from the t-distribution table or use the p-value approach to decide if we reject or fail to reject H_0 . If the p-value is less than the significance level (commonly 0.05), then there is statistically significant evidence to reject the null hypothesis.

06

Considering the Study's Context

Consider that the study's sample consists of college athletes only, which may not be representative of the general population. This context suggests that results should be cautiously applied beyond the studied group.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Sample T-Test

A two-sample t-test is a statistical method used to determine if there is a significant difference between the means of two independent groups. In our exercise, researchers used this test to compare hippocampal volume between two groups of football players: those with a concussion history and those without.

In this context, the two-sample t-test requires us to calculate the test statistic, ‘t’, using the formula:

• Calculate the difference between the two sample means.
• Divide this difference by the standard error of the mean difference, which accounts for the variability and size of each group.

This test helps assess whether any observed differences are likely due to chance or if they reflect a real difference in brain size resulting from concussions.

Null Hypothesis

The null hypothesis (\(H_0\)) serves as the starting assumption in hypothesis testing. It posits that there is no effect or difference between groups. For our specific inquiry into the effect of concussions on brain size, the null hypothesis asserts that \(\mu_1 = \mu_2\), meaning the average hippocampal volume is the same for both football players with and without a concussion history.

It is essential to test the validity of this claim before making any conclusions.

• If data shows sufficient evidence to refute the null hypothesis, it indicates there is likely a real difference.
• If not, we fail to reject the null hypothesis, implying no strong evidence of a difference.

Judging the null hypothesis gives a baseline measure against which the alternative is tested.

Alternative Hypothesis

The alternative hypothesis (\(H_a\)) is the statement that researchers aim to provide evidence for. It is formulated in direct contrast to the null hypothesis, suggesting that there is indeed a difference in the means we are analyzing. For the football players' brain size study, the alternative hypothesis is that \(\mu_1 eq \mu_2\).

This decision is vital as it guides the direction of testing. When the null hypothesis is rejected, the alternative hypothesis becomes the accepted explanation, showing a notable impact or change:

• Researchers seek to validate \(H_a\), especially if the study context supports a reasonable cause to expect such a result.
• The acceptance of the alternative hypothesis can indicate meaningful differences, such as changes in hippocampal volume due to concussion history.

Statistical Significance

Statistical significance is a key concept determining if the results of a study are likely meaningful. It is measured using the p-value, often compared against a significance level (commonly set at 0.05). When the p-value is below this threshold, the results are considered significant.

In the context of this study, researchers used statistical significance to decide whether to reject the null hypothesis:

• A p-value less than 0.05 suggests the difference in brain sizes between the two groups is unlikely due to random chance.
• This means a real effect or difference exists, possibly attributed to historical concussions.

Statistically significant results reinforce the conclusion that researchers can reject \(H_0\) with confidence, suggesting a genuine difference in outcomes examined.