please explain step by step The professor decides to use the Tukey HSD method to test all possible pairwise contrasts. To do so, thefollowing table of q; statistics needs to be completed. (Note: we often create a table of differences andcompare those to HSD. In this case, we need a table of q statistics, which we will compare to qcritical) Forthis presentation, the groups will be ordered from smallest to largest means. (In other words, Group Ashould be the group with the smallest mean.) First, indicate the appropriate order for the groups (use thegroup number). Then fill in the table with values rounded to 3 decimal places. (Furthermore, even thoughthe sign does not matter, report all comparison statistics as positive values.)Comparison GroupsGroups(ascending order)Group AGroup BGroup CGroup DGroup A == 3Group B =Group C =Group D =What is the critical value of the studentized range for the Tukey HSD test (a0.01)?CcritMark all pairwise comparisons that are statistically significant A professor gives an exam to a class with four recitation sections. Each section has 11 students. Theprofessor has no reason to assume that these exam scores would not be independent and normallydistributed with equal variance. However, it is unknown whether or not the section choice (different TASand different days of the week) has any relationship with how students performed on the test.Group-1Group-2Group-3Group-471.564.183.377.575.268.963.47959.481.655.273.750.177.755.575.473.284.46976.641.569.844.874.467.678.56562.568.665.964.665.260.878.871.898.259.170.261.972.684.869.346.370.6

Given Information : A professor gives an exam to a class with four recitation sections. Each section has 11 students . The professor as no reason to assume that these exam scores would not be independent and normally distributed with equal variance . However , it is unknown whether or not the section choice (Different TAS and different days o the week) has any relationship with how students performed the test . Input data has k = 4 treatments . Treatment → A B C D Input Data → 71.5 64.1 83.3 77.5 75.2 68.9 63.4 79 59.4 81.6 55.2 73.7 50.1 77.7 55.5 75.4 73.2 84.4 69 76.6 41.5 69.8 44.8 74.4 67.6 48.5 65 62.5 68.6 65.9 64.6 65.2 60.8 78.8 71.8 98.2 59.1 70.2 61.9 72.6 84.8 69.3 46.3 70.6 Descriptive statistics for k = 4 independent treatments : Treatment → A B C D Pooled Total observations N 11 11 11 11 44 sum ∑xi 711.8 779.2 680.8 825.7 2,997.50 mean ¯x 64.7091 70.8364 61.8909 75.0636 68.125 sum of squares ∑x2i 47,542.36 56,182.30 43,382.48 62,822.67 2,09,929.81 sample variance s2 148.2429 98.6605 124.7149 84.2625 133.1424 sample std. dev. S 12.1755 9.9328 11.1676 9.1795 11.5387 std. dev. of mean SE¯x 3.6711 2.9949 3.3672 2.7677 1.7395 One - way ANOVA for k = 4 independent treatments : source sum of Squares degrees of freedom v mean square MS F statistic p-value treatment 1,166.31 3 388.7711 3.4112 0.0265 error 4,558.81 40 113.9702 total 5,725.12 43 Conclusion from Anova: The p-value corresponding to the F-statistic of one-way ANOVA is lower than 0.05 which strongly suggests that one or more pairs of treatments are significantly different. You have k=4 treatments, for which we shall apply Tukey's HSD test to each of the 6 pairs to pinpoint which of them exhibits statistically significant difference. Tukey HSD Test: First establish the critical value of the Tukey-Kramer HSD Q statistic based on the k=4 treatments and ν=40 degrees of freedom for the error term, for significance level α= 0.01 and 0.05 (p-values) in the Studentized Range distribution. We obtain these ctitical values for Q, for α of 0.01 and 0.05, as Qcriticalα=0.01,k=4,ν=40 = 4.6953 and Qcriticalα=0.05,k=4,ν=40 = 3.7909, respectively. These critical values may be verified at several published tables of the inverse Studentized Range distribution. Next, we establish a Tukey test statistic from our sample columns to compare with the appropriate critical value of the studentized range distribution. We take the Tukey-Kramer confidence limits and make simplifying algebraic transformation. We calculate a parameter for each pair of columns being compared, which we loosely call here as the Tukey-Kramer HSD Q-statistic, or simply the Tukey HSD Q-statistic, as: Qi,j=x¯i-x¯jsi,j where the denominator in the above expression is: si,j=σ^∈Hi,j i,j=1,2...k ; i≠j The quantity Hi,j is the harmonic mean of the number of observations in columns labeled i and j. Note that when the sample sizes in the columns are equal, then their harmonic mean is simply the common sample size. When the sample sizes of columns in a pair being compared are different, the harmonic mean lies somewhere in-between the two sample sizes. The relvant harmonic mean is required for applying the Tukey-Kramer procedure for columns with unequal sample sizes. The quantity σ^ϵ = 10.6757 is the square root of the Mean Square Error = 113.9702 determined in the precursor one-way ANOVA procedure. Note that ^σϵ is same across all pairs being compared. The only factor that varies across pairs in the computation of si,j=σ^∈Hi,j is the denominator, which is the harmonic mean of the sample sizes being compared. The test of whether the NIST Tukey-Kramer confidence interval includes zero is equivalent to evaluating whether Qi,j>Qcritical, the latter determined according to the desired level of significance α (p-value), the number of treatments kk and the degrees of freedom for error νν, as described above. post-hoc Tukey HSD Test Calculator results: k=4 treatmentsdegrees of freedom for the error term ν=40ν=40Critical values of the Studentized Range Q statistic: Qcriticalα=0.01,k=4,ν=40 = 4.6953 Qcriticalα=0.05,k=4,ν=40 = 3.7909 We present below color coded results (red for insignificant, green for significant) of evaluating whether Qi,j>Qcritical for all relevant pairs of treatments. In addition, we also present the significance (p-value) of the observed Q-statistic Qi,j. The algorithm used here to calculate the critical values of the studentized range distribution, as well as p-values corresponding to an observed value of Qi,j. Tukey HSD results treatments Tukey HSD Tukey HSD Tukey HSD pair Q statistic p-value inference A vs B 1.9036 0.5360035 insignificant A vs C 0.8755 0.8999947 insignificant A vs D 3.2169 0.1213909 insignificant B vs C 2.7791 0.2183217 insignificant B vs D 1.3133 0.7667118 insignificant C vs D 4.0924 0.0300269 * p<0.05