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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 ADHD and a representative sample of 139 children without ADHD. Summary values for total cerebral volume (in milliliters) are given in the following table: $$ \begin{array}{lccc} & n & \bar{x} & s \\ \hline \text { Children with ADHD } & 152 & 1,059.4 & 117.5 \\ \text { Children Without ADHD } & 139 & 1,104.5 & 111.3 \end{array} $$ Use a \(95 \%\) confidence interval to estimate the differ- ence in mean brain volume for children with and without ADHD.

Step by step solution

01

Identify the given values

First, identify the summary statistics given for each group: \( n_1 = 152, \bar{x_1} = 1059.4, s_1 = 117.5 \) for the ADHD group and \( n_2 = 139, \bar{x_2} = 1104.5, s_2 = 111.3 \) for the non-ADHD group.

02

Calculate the standard error for the difference in means

The formula to compute Standard Error (SE) for difference of two means is \[ SE = \sqrt{\left(\frac{{s_1}^2}{n_1}\right) + \left(\frac{{s_2}^2}{n_2}\right)} \] Substituting the given values, you will obtain the standard error.

03

Calculate the 95% confidence interval for the difference in means

The formula for the 95% confidence interval (\( CI \)) for the difference in two means is \[ CI = (\bar{x_1} - \bar{x_2}) \pm z \cdot SE \] where \( z \) is the z-score corresponding to the desired confidence level (1.96 for 95% confidence level). With the calculated standard error and the given mean values, you will arrive at the confidence interval for the difference in mean brain volume for children with and without ADHD.

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Key Concepts

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

ADHD Brain Volume Research

The relevance of research into the brain volume of children with ADHD lies in its potential to enhance our understanding of ADHD's neurological underpinnings. Brain imaging studies, like the one cited in the journal of the American Medical Association, aim to determine if structural differences exist between individuals with and without ADHD. This can offer insights into how such differences may affect behavior and cognitive functions associated with the disorder. High-quality research in this field can lead to more effective diagnostic methods and interventions, tailoring treatments to specific neurological profiles.

Standard Error Calculation

Understanding the concept of standard error (SE) is critical when interpreting data from studies like the ADHD brain volume research. SE measures the amount of variability in sample statistics from one sample to another if repeated samples were taken. To compute the standard error for the difference in means between two groups, you use the formula: \[ SE = \sqrt{\left(\frac{{s_1}^2}{n_1}\right) + \left(\frac{{s_2}^2}{n_2}\right)} \] This allows researchers to estimate the uncertainty surrounding the difference calculated from the sample means. A lower SE indicates that the sample mean is a more precise reflection of the true population mean.

Difference in Means

In research studies that compare two groups, a common statistic of interest is the difference in means. This measures the discrepancy between the average values of the two groups, providing a simple yet powerful means of assessing the contrast in their central tendencies. For example, in the ADHD brain volume study, the difference in mean brain volume between children with ADHD and those without is a focal outcome that could demonstrate a potential impact of the condition on brain development. By comparing the mean cerebral volumes, \( \bar{x_1} - \bar{x_2} \), we capture a snapshot of how the two groups differ on average.

Statistical Significance

Statistical significance is a cornerstone concept in hypothesis testing, helping us determine if our findings are likely to reflect true differences in the population or merely due to random chance. When calculating a 95% confidence interval for the difference in means, as seen in the ADHD research, we're essentially saying there's a 95% chance that the interval contains the true difference in brain volumes. The '95%' is a level of confidence selected to mitigate the likelihood of false-positive results while still remaining sensitive to true effects. If the confidence interval does not include zero, this suggests that the observed difference is statistically significant and unlikely to have occurred because of random variation alone.