Sleep apnea is a sleep disorder characterized by pauses in breathing during sleep. Children with sleep apnea have behavior problems, including hyperactivity, inattention, and aggression, as well as impaired learning and diminished academic performance. The removal of tonsils and adenoids that are enlarged, causing the obstruction of the airways, is one of the most common treatments for pediatric sleep apnea. A clinical psychologist studies the effects of tonsillectomy and adenoidectomy on hyperactive behavior. Her quasi-experiment includes three groups of 11 children. The first group of children does not have sleep apnea, the second group has untreated sleep apnea, and the third group has sleep apnea treated by tonsillectomies and adenoidectomies. Hyperactivity was measured using teacher reports on the Conners Rating Scale. The sample means and sums of squared deviations of the scores for each of the three groups are presented in the table that follows. Group Sample Mean Sum of Squares No sleep apnea 0.42 0.3240 Untreated sleep apnea 0.56 0.4410 Treated sleep apnea 0.30 0.2250   After collecting the data, the clinical psychologist analyzes the data using an ANOVA. The results of her analysis are presented in the ANOVA table that follows. ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F Between Treatments 0.3729 2 0.1865 5.65 Within Treatments 0.9900 30 0.0330   Total 1.3629 32       The critical value of F when α = 0.05 is 3.316, meaning the critical region consists of all F-ratios greater than 3.316. The F-ratio is greater than this critical value, so you know that at least one difference exists among the treatments. Since more than two groups are involved, the psychologist is interested in determining which groups are different. The Scheffe test will be used to evaluate the pairs. Call the no sleep apnea group A, the untreated sleep apnea group B, and the treated sleep apnea group C. Start with the calculations you will need to evaluate the difference between the no sleep apnea group (A) and the untreated sleep apnea group (B). The SSbetween ABbetween AB is    . The FA versus BA versus B is    .   (Hint: Recall that you can use the sample size for each treatment (n = 11) and the treatment mean to compute each treatment total (T).) At α = 0.05, the psychologist    conclude that the population means for children without sleep apnea and children with untreated sleep apnea differ.   Next evaluate the difference between the no sleep apnea group (A) and the treated sleep apnea group (C). The SSbetween ACbetween AC is    . The FA versus CA versus C is    .   At α = 0.05, the psychologist    conclude that the population means for children without sleep apnea and children with treated sleep apnea differ.   Next calculate the values necessary to evaluate the difference between the untreated sleep apnea group (B) and the treated sleep apnea group (C). The SSbetween BCbetween BC is    . The FB versus CB versus C is    .   At α = 0.05, the psychologist    conclude that the population means for children with untreated sleep apnea and children with treated sleep apnea differ.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...