Sleep apnea is a disorder characterized by pauses in breathing during sleep. Children who suffer from untreated sleep apnea often have behavior problems, including hyperactivity, inattention, and aggression. A common treatment for pediatric sleep apnea is the surgical removal of enlarged tonsils and adenoids that are obstructing the airways.Suppose researchers at a sleep clinic are interested in the effect of surgical treatment for pediatric sleep apnea on aggressive behavior. They study 11 children without sleep apnea, 11 children with untreated sleep apnea, and 11 children who have been surgically treated for sleep apnea. Aggression is measured using the Conners Rating Scales.The sample means and sums of squares of the scores for each of the three groups are presented in the following table.GroupSample MeanSum of SquaresNo Sleep Apnea0.590.3240Untreated Sleep Apnea0.450.4410Treated Sleep Apnea0.310.2250 The researchers perform an analysis of variance (ANOVA) at α = 0.01 to test the hypothesis that the treatment means are equal. The results are presented in the following ANOVA table.ANOVA TableSource of VariationSum of SquaresDegrees of FreedomMean SquareFBetween Treatments0.431220.21566.53Within Treatments0.9900300.0330 Total1.421232 The ANOVA yielded a significant F statistic, so the null hypothesis is rejected. Since there are more than two groups, the researchers are interested in determining which groups are different. The Tukey’s Honestly Significant Difference (HSD) test will be used to evaluate the pairs. First, use the table given below to determine the appropriate value of q at α = 0.01. The q value for this problem is . The Studentized Range Statistic (q)df for Error Term23202.953.58 4.024.64242.923.53 3.964.55302.893.49 3.894.45402.863.44 3.824.37602.833.40 3.764.28The top value is α = .05; the bottom (bold) value is α = .01. The number of treatments is listed across. The df for the error term is in the left column, where the “error term” is another name for the within-treatments variance. Now, use the q value to calculate Tukey’s HSD (round to two decimal places). Tukey’s HSD is . Thus, the mean difference between any two samples must be at least to be significant. The researchers conclude that the population means for children without sleep apnea and children with untreated sleep apnea differ. They conclude that the population means for children without sleep apnea and children with treated sleep apnea differ. They conclude that the population means for children with untreated sleep apnea and children with treated sleep apnea differ.

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Description: Sleep apnea is a disorder characterized by pauses in breathing during sleep. Children who suffer from untreated sleep apnea often have behavior problems, including hyperactivity, inattention, and aggression. A common treatment for pediatric sleep apn

Test hypotheses: The 3 groups of study are No Sleep Apnea, Untreated Sleep Apnea and Treated Sleep…

Answered: Sleep apnea is a disorder characterized…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter10: Matrices

Section10.EA: Extended Application Contagion

Problem 2EA

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Sleep apnea is a disorder characterized by pauses in breathing during sleep. Children who suffer from untreated sleep apnea often have behavior problems, including hyperactivity, inattention, and aggression. A common treatment for pediatric sleep apnea is the surgical removal of enlarged tonsils and adenoids that are obstructing the airways.

Suppose researchers at a sleep clinic are interested in the effect of surgical treatment for pediatric sleep apnea on aggressive behavior. They study 11 children without sleep apnea, 11 children with untreated sleep apnea, and 11 children who have been surgically treated for sleep apnea. Aggression is measured using the Conners Rating Scales.

The sample means and sums of squares of the scores for each of the three groups are presented in the following table.

Group	Sample Mean	Sum of Squares
No Sleep Apnea	0.59	0.3240
Untreated Sleep Apnea	0.45	0.4410
Treated Sleep Apnea	0.31	0.2250

The researchers perform an analysis of variance (ANOVA) at α = 0.01 to test the hypothesis that the treatment means are equal. The results are presented in the following ANOVA table.

ANOVA Table

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F
Between Treatments	0.4312	2	0.2156	6.53
Within Treatments	0.9900	30	0.0330
Total	1.4212	32

The ANOVA yielded a significant F statistic, so the null hypothesis is rejected. Since there are more than two groups, the researchers are interested in determining which groups are different. The Tukey’s Honestly Significant Difference (HSD) test will be used to evaluate the pairs. First, use the table given below to determine the appropriate value of q at α = 0.01. The q value for this problem is .

The Studentized Range Statistic (q)

df for Error Term	2	3
20	2.95	3.58
	4.02	4.64
24	2.92	3.53
	3.96	4.55
30	2.89	3.49
	3.89	4.45
40	2.86	3.44
	3.82	4.37
60	2.83	3.40
	3.76	4.28

The top value is α = .05; the bottom (bold) value is α = .01. The number of treatments is listed across. The df for the error term is in the left column, where the “error term” is another name for the within-treatments variance.

Now, use the q value to calculate Tukey’s HSD (round to two decimal places). Tukey’s HSD is . Thus, the mean difference between any two samples must be at least to be significant.

The researchers conclude that the population means for children without sleep apnea and children with untreated sleep apnea differ.

They conclude that the population means for children without sleep apnea and children with treated sleep apnea differ.

They 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 + ...

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