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Do children diagnosed with attention deficit/ hyperactivity disorder (ADHD) have smaller brains than children without this condition? This question was the topic of a research study described in the paper "Developmental Trajectories of Brain Volume Abnormalities in Children and Adolescents with Attention Deficit/Hyperactivity Disorder" (Journal of the American Medical Association [2002]: \(1740-1747\) ). Brain scans were completed for a representative sample of 152 children with ADHI) and a representative sample of 139 children without \(A D H D\). Summary values for total cerebral volume (in cubic milliliters) are given in the following table: Is there convincing evidence that the mean brain volume for children with ADHD is smuller than the mean for children without ADHD? Test the relevant hypotheses using a 0.05 level of significance.

Step by step solution

01

State the hypothesis

The null hypothesis (H0) states that there is no difference between the mean brain volume of children with ADHD and those without ADHD. The alternative hypothesis (Ha) states that the mean brain volume of children with ADHD is smaller than that of children without ADHD. \(H_0: \mu_a = \mu_b\) \(H_a: \mu_a < \mu_b\) Where: - \(\mu_a\) is the mean brain volume of children with ADHD - \(\mu_b\) is the mean brain volume of children without ADHD

02

Calculate the test statistics

To test our hypothesis, we will perform a two-sample t-test. We have the following data: Children with ADHD (group A): - Number of subjects (n_a): 152 - Mean brain volume (x̄_a): given in the problem Children without ADHD (group B): - Number of subjects (n_b): 139 - Mean brain volume (x̄_b): given in the problem First, we need to calculate the pooled variance and standard deviation. \(s_p^2 = \frac{(n_a - 1)s_a^2 + (n_b - 1)s_b^2}{n_a + n_b - 2}\) Where: - \(s_a^2\) is the variance of group A (ADHD) - \(s_b^2\) is the variance of group B (non-ADHD) We also need the pooled standard deviation: \(s_p = \sqrt{s_p^2}\) Next, we calculate the t-test statistic: \(t = \frac{(x̄_a - x̄_b) - (\mu_a - \mu_b)}{s_p \sqrt{\frac{1}{n_a}+\frac{1}{n_b}}}\) Using the given data, compute the t-test statistic.

03

Determine the critical value

Our level of significance is 0.05. Since this is a one-tailed test (we are only interested if the mean brain volume of ADHD children is smaller than non-ADHD children), we use a one-tailed t-distribution table to find the critical value (t_critical) with the relevant degrees of freedom (df). Degrees of freedom (df) is calculated as: \(df = n_a + n_b - 2\) Find the t_critical value in the table, given the level of significance and degrees of freedom.

04

Compare the test statistic and critical value

Now, we need to compare the calculated t-test statistic with the critical value: - If t ≤ t_critical, we reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha). - If t > t_critical, we fail to reject the null hypothesis (H0).

05

Draw conclusions

Based on our comparison of the t-test statistic and the critical value, we can determine if there is convincing evidence that the mean brain volume of children with ADHD is smaller than that of children without ADHD. If we reject H0, then we can conclude that there is evidence to support the claim that ADHD children have smaller brain volumes. If we fail to reject H0, then we cannot conclude that ADHD children have smaller brain volumes compared to non-ADHD children.

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Key Concepts

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

Null Hypothesis

In any statistical test, especially the two-sample t-test we're discussing, the null hypothesis (denoted as \( H_0 \)) is a fundamental concept. It represents the default position that there is no effect or no difference between the groups being compared. In our exercise, the null hypothesis states that the mean brain volume of children with ADHD is equal to that of children without ADHD.

This can be mathematically expressed as:
\[ H_0: \mu_a = \mu_b \]
Where:

• \( \mu_a \) is the mean brain volume of children with ADHD
• \( \mu_b \) is the mean brain volume of children without ADHD

The null hypothesis is what we set out to test and either reject or fail to reject, based on the evidence provided by our data. It is crucial to remember that failing to reject the null hypothesis does not prove it true; it merely indicates insufficient evidence to declare a difference.

Alternative Hypothesis

On the other hand, the alternative hypothesis (denoted as \( H_a \)) proposes that there is a difference between the two groups. For our study, the alternative hypothesis suggests that the mean brain volume of children with ADHD is smaller than that of children without ADHD. This is what we are aiming to find evidence for.

Mathematically, this is expressed as:
\[ H_a: \mu_a < \mu_b \]
This alternative hypothesis is directional, as it specifies not just a difference but a specific direction of the difference (smaller brain volumes in the ADHD group). The choice between a directional or non-directional hypothesis can impact the statistical testing process, particularly regarding the critical values used.

Levels of Significance

The level of significance, often denoted as \( \alpha \), is a crucial parameter in hypothesis testing. It determines the threshold at which we will reject the null hypothesis. In our scenario, the level of significance chosen is 0.05, or 5%.

This means there is a 5% risk that we conclude there is a difference when there is none (Type I error). It's a balance point between being too cautious and too liberal with our findings.

The level of significance helps define the critical region of our test statistic distribution. If the computed test statistic falls into this region, it supports rejecting the null hypothesis. Lower levels of significance, like 0.01, require stronger evidence to reject the null, while higher levels, like 0.10, require less. The choice of \( \alpha \) can influence test outcomes and interpretations.

Degrees of Freedom

Degrees of freedom (df) are a concept that comes up frequently in statistical analyses and are critical in determining test statistics' distributions. In our two-sample t-test, the degrees of freedom are calculated as the sum of the sizes of the two groups minus two. That is,

\[ df = n_a + n_b - 2 \]
where \( n_a \) and \( n_b \) are the number of subjects in groups A and B, respectively.

Degrees of freedom represent the number of values in the final calculation of a statistic that are free to vary. They help in determining the specific shape of the t-distribution we will use for hypothesis testing. A larger number of degrees of freedom generally makes the t-distribution closer to a normal distribution, which typically allows for more robust statistical analyses.