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In: Statistics and Probability

11. The Scheffe test Sleep apnea is a sleep disorder characterized by pauses in breathing during...

11. The Scheffe test

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 aggressive behavior. Her quasi-experiment includes three groups of 9 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. Aggression was measured using parent 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 0.26 0.2888
Untreated sleep apnea 0.62 0.3872
Treated sleep apnea 0.33 0.2312

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 0.6561 2 0.3281 8.66
Within Treatments 0.9072 24 0.0378
Total 1.5633 26

The critical value of F when α = 0.01 is 5.614, meaning the critical region consists of all F-ratios greater than 5.614. 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 = 9) and the treatment mean to compute each treatment total (T).)

At α = 0.01, 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 = 0.01, 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 = 0.01, the psychologist   conclude that the population means for children with untreated sleep apnea and children with treated sleep apnea differ.

Solutions

Expert Solution

group sample size sample mean
No sleep Apnea (A) 9 0.26
Untreated Sleep Apnea (B) 9 0.62
Treated Sleep Apnea ( C ) 9 0.33
Df between= 2
MSwithin= 0.0378
Critical F = 5.614

grand mean AB, x̅̅ = Σni*x̅i/Σni =     (9*0.26+9*0.62)/(9+9)=       0.44                          
SS(between AB) = Σn( x̅ - x̅̅)² =    9*(0.26-0.44)²+9*(0.62-0.44)²=       0.5832                          
                                      
MS(between AB)=SS(between AB)/df between=   0.5832/2=   0.2916                              
F (A vs B) =    0.2916/0.0378=   7.7143                              
since, F stat > critical F value, Reject the null hypothesis                                      
                                      
The SSbetween AB is    0.5832                                  
The F (A versus B) is    7.7143                                  
the psychologist    can   conclude that the population means for children without sleep apnea and children with untreated sleep apnea differ                             
============================================================================================                                      
grand mean AC, x̅̅ = Σni*x̅i/Σni =     (9*0.26+9*0.33)/(9+9)=       0.295                          
SS(between AC) = Σn( x̅ - x̅̅)² =    9*(0.26-0.295)²+9*(0.33-0.295)²=       0.0221                          
                                      
MS(between AC)=SS(between AC)/df between=   0.0221/2=   0.0110                              
F (A vs C) =    0.011/0.0378=   0.2917                              
Since, F stat < critical F value , Do not reject null hypothesis                                      
                                      
The SSbetween AC is    0.0221                                  
The F (A versus C) is    0.2917                                  
the psychologist    cannot   conclude that the population means for children without sleep apnea and children with treated sleep apnea differ                              
=================================================================================================                                      
grand mean BC, x̅̅ = Σni*x̅i/Σni =     (9*0.62+9*0.33)/(9+9)=       0.475                          
SS(between BC) = Σn( x̅ - x̅̅)² =    9*(0.62-0.475)²+9*(0.33-0.475)²=       0.3785                          
                                      
MS(between BC)=SS(between BC)/df between=   0.3785/2=   0.1892                              
F (B vs C) =    0.1892/0.0378=   5.0060                              
Since, F stat < critical F value , Do not reject null hypothesis                                      
                                      
The SSbetween BC is    0.3785                                  
The F (B versus C) is    5.0060                                  
the psychologist    cannot   conclude that the population means for children with untreated sleep apnea and children with treated sleep apnea differ                              
                                      

                                      
                                      
                                      


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