Tukey's HSD test 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.

Let X be a random variable with moment generating function mx (t) e -84 24 3 3 6 a) Use the derivatives of my (t) to find the mean and variance of X. b) Find the probability mass function of X and use it to check your results for the mean and variance you derived in a).Let X ~ H[1{],4,3}. 1. compute the values of the itznroidaiolitvr mass function of X. 2. Use the definition Mxit] = E [entXJ = summation from x {ehtxmxi} of moment generating function to derive Mxit}. 3. Find the mean and variance of K using the moment generating function. Compare these value with known formulas. Problem 1 Let X denote a geometric random variable with parameter p E (0, 1) such that P(X = n) = (1 - p)p"-1, for n = 1, 2, . ... (1) Derive and calculate the mean of X directly. (2) Derive and calculate the variance of X directly. (3) Derive the moment generating function (Z-transform) of X. (4) Using the moment generating function, derive the mean and the variance of X.10. Tukey's HSD test Sleep apnea is a disorder characterized by pauses in breathing during sleep. Children who sufferfrom 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 inattentive behavior. Ther-r studs.r nine children Iwithout sleep apnea, nine children with untreated sleep apnea. and nine children who have been surgically treated for sleep apnea. Inattentiveness is measured using the Conners Rating Scales. The sample means and sums of squares ofthe scores for each of the three groups are presented in the following table. Group Sample Mean Sum of Squares No Sleep Apnea 0.27 0.1568 Untreated Sleep Apnea 0.54 0.2592 Treated Sleep Apnea 0.32 0.1800 The researchers perform an analysis of variance {ANOVA} at o = 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 0.3714 2 0.1857 7.49 Within Treatments 0.5960 24 0.0248 Total 0.9674 26 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 a = 0.01. The q value for this problem isThe 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 a = 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 a = .05; the bottom (bold) value is a = .01. The number of treatments is listed across. The of 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