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. 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.
n x s
Children with ADHD 152 1,058.7 116.7
Children without ADHD 139 1,104.2 111.6
Is there convincing evidence that the mean brain volume for children with ADHD is smaller than the mean
for children without ADHD? Test the relevant hypotheses using a 0.05 level of significance. (Use 𝜇 for
children with ADHD and 𝜇 for children without ADHD.)
State the appropriate null and alternative hypotheses (in millimeters).
Find the test statistic. (Round your answer to two decimal places.)
Use technology to find the P-value. (Round your answer to four decimal places.)
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Null hypothesis: 𝜇_1 = 𝜇_2 (mean brain volume for children with ADHD is equal to mean brain volume for children without ADHD)
Alternative hypothesis: 𝜇_1 < 𝜇_2 (mean brain volume for children with ADHD is smaller than mean brain volume for children without ADHD)
The test statistic can be calculated using the formula:
t = (x_1 - x_2) / sqrt(s_1^2/n_1 + s_2^2/n_2)
where x_1 and x_2 are the sample means, s_1 and s_2 are the sample standard deviations, and n_1 and n_2 are the sample sizes.
Using the given information, the test statistic can be calculated as:
t = (1058.7 - 1104.2) / sqrt((116.7^2/152) + (111.6^2/139))
t ≈ -4.10 (rounded to two decimal places)
Using technology to find the P-value, the P-value is approximately 0.0001 (rounded to four decimal places).
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