Question

Attention deficit hyperactivity disorder (ADHD) in young children presents significant challenges to the children, their parents, and their classroom teachers. Stimulant medication can effectively treat 60% to 70% of youth with ADHD. Yet many parents seek alternative therapies, and Hypericum perforatum (St John's wort) is 1 of the top 3 alternative therapies used.
In a recent study in the Journal of the American Medical Association, Hypericum perforatum (St John's Wort) for Attention-Deficit/Hyperactivity Disorder in Children and Adolescents, 2008;299(22):2633-2641, the authors conducted a randomized, double-blind, placebo-controlled trial to determine the efficacy and safety of St. John's wort for the treatment of ADHD in children.
After careful pre-screening, 54 children were randomly assigned to two treatment groups; 27 children were assigned to receive a placebo treatment and 27 children were assigned to receive a treatment with St. John's wort.
A measure of "inattentiveness" was used to evaluate the effect of the treatments. At the beginning of the experiment, the placebo group and treatment group had the same inattentiveness score. After 4 weeks of the experiment, the data in the table below were obtained (a lower inattentiveness score indicates a longer attention span):

ST. JOHN'S WORT vs PLACEBO STUDY

Placebo Group	St. John's Wort Group
x1 = 14.0	x2 = 13.4
s1 = 3.87	s2 = 3.22
n1 = 27	n2 = 27

Question 1. Calculate a 99% confidence interval for the difference μ₁ - μ₂, where μ₁ is the mean inattentiveness score after 4 weeks for all ADHD children who receive a placebo, and μ₂ is the mean inattentiveness score after 4 weeks for all ADHD children who receive St. John's wort.

  lower bound of interval
  upper bound of interval

Question 2.Select the choice below that correctly interprets the interval in question 1.

I am 99% confident that the average change of the inattentiveness score for children in this study is in the interval.

I am 99% confident that the interval in question 1 captures the true mean difference μ₁ - μ₂; based on this interval it does not appear that St. John's wort is an effective treatment for ADHD.   

I am 99% confident that if a child with ADHD is given St John's wort, the child's inattentiveness score will change by an amount in the interval.

If St. John's wort is tested on another sample of ADHD children, there is a 99% chance that their average inattentiveness score will change by an amount in the interval.

The inattentiveness score of 99% of the children studied changed by an amount in the interval.

The probability that μ₁ - μ₂ is in the interval is 0.99.

Answer 1

Q1:

For Placebo Group:

x̅1 = 14, s1 = 3.87, n1 = 27

For St. John's Wort Group:

x̅2 = 13.4, s2 = 3.22, n2 = 27

99% Confidence interval :

At α = 0.01 and df = n1+n2-2 = 52, two tailed critical value, t-crit = T.INV.2T(0.01, 52) = 2.674

S²p = ((n1-1)*s1² + (n2-1)*s2² )/(n1+n2-2) = ((27-1)*3.87² + (27-1)*3.22²) / (27+27-2) = 12.6727

Lower Bound = (x̅1 - x̅2) - t-crit*√(S²p*(1/n1 +1/n2)) = (14 - 13.4) - 2.674*√(12.6727*(1/27 + 1/27)) = -1.9905

Upper Bound = (x̅1 - x̅2) + t-crit*√(S²p*(1/n1 +1/n2)) = (14 - 13.4) + 2.674*√(12.6727*(1/27 + 1/27)) = 3.1905

-1.9905 < µ1 - µ2 < 3.1905

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Q2:

Answer:

I am 99% confident that the interval in question 1 captures the true mean difference μ₁ - μ₂; based on this interval it does not appear that St. John's wort is an effective treatment for ADHD.