Problem V:(40pts) 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 150 children with ADHD and 150 children of similar age without ADHD. Summary values for total cerebral volume (in milliliters) are given in the following table: Children with ADHD Children without ADHD Sample Size Sample cerebral volume Sample standard deviation 150 1059.4 117.5 150 1104.5 111.3 1. Find a 95% confidence interval for the mean difference. 2. Answer the question: “Do children diagnosed with attention deficit/hyperactivity dis- order (ADHD) have smaller brains than children without this condition?” using sig- nificance level a = 0.02 3. Find the confidence level if we want to have a confidence level of length 30 for the difference in mean . (with data above) 4. If the estimate of the difference in mean is to be correct to within 30 milliliters with probability 0.95, how many children needs to be surveyed (assume ni = n2).
The correct answer and explanation is:
Results:
- 95% Confidence Interval for the Mean Difference:
[
(-71.21, -18.99)
]
This means that the average cerebral volume of children with ADHD is estimated to be between 18.99 and 71.21 milliliters smaller than that of children without ADHD. - Hypothesis Test (α = 0.02):
- Null Hypothesis: ( H_0: \mu_1 = \mu_2 ) (No difference in brain volume)
- Alternative Hypothesis: ( H_A: \mu_1 < \mu_2 ) (ADHD children have smaller brains)
- p-value = 0.00083
Since the p-value (0.00083) < 0.02, we reject the null hypothesis. This means there is significant evidence that children with ADHD have smaller brains than those without ADHD.
- Confidence Level for a Confidence Interval of Length 30:
- The computed confidence level is 25.63%, which is very low. This suggests that a confidence interval with a length of 30 would not be reliable unless we increase the sample size.
- Required Sample Size to Ensure an Estimate Within 30 ml (95% Probability):
- 224 children per group (total = 448 children) are needed to ensure that the estimated mean difference is within 30 milliliters with 95% probability.
Now, I will generate a visualization to illustrate these results.
The plot above visually represents the 95% confidence interval for the mean difference in brain volume between children with ADHD and those without. The red error bar shows the confidence interval range, with the lower bound at -71.21 mL and the upper bound at -18.99 mL. Since this interval does not include zero, it supports the conclusion that children with ADHD have smaller brains on average than those without the condition.