Air traffic controllers perform the vital function of regulating the traffic of

Question

Air traffic controllers perform the vital function of regulating the traffic of passenger planes. Frequently, air traffic controllers work long hours with little sleep. Researchers wanted to test their ability to make basic decisions as they become increasingly sleep deprived. To test their abilities, a sample of 6 air traffic controllers is selected and given a decision-making skills test following 12-hour, 24-hour, and 48-hour sleep deprivation. Higher scores indicate better decision-making skills. The table lists the hypothetical results of this study. Sleep Deprivation 12 Hours 24 Hours 48 Hours 22 18 16 19 21 20 34 22 21 27 20 14 22 15 16 20 20 15 (a) Complete the F-table. (Round your answers to two decimal places.) Source of Variation SS df MS Fobt Between groups Between persons Within groups (error) Total (b) Compute a Bonferroni procedure and interpret the results. (Assume experimentwise alpha equal to 0.05. Select all that apply.) There is a significant difference in decision making for the 12-hour and 48-hour sleep deprivation conditions. There are no significant differences between any of the groups. There is a significant difference in decision making for the 12-hour and 24-hour sleep deprivation conditions. There is a significant difference in decision making for the 24-hour and 48-hour sleep deprivation conditions.

Explanation / Answer

a. In order to complete the ANOVA table, do the following calculations.

SST=Sigma Xi^2-NXbar^2=(3614+2274+1774)-18(20.11)^2=382.5822

SSB=Sigma Nk(Xk bar-Xbar)^2=6(24-20.11)^2+6(19.33-20.11)^2+6(17-20.11)^2=152.4756

SSW=SST-SSB=230.1066

df(Within)=N-k=18-3=15; df(Between)=k-1=3-1=2, where, k i snumber of groups and N is sample size.

MSW=SSW/dfw=15.34044; MSB=SSB/dfB=76.2378

F=MSB/MSW=4.97, p value=0.022

The p value is less than 0.05, reject null hypothesis [H0:mu1=mu2=mu3] and conclude that at least two mean hours differ.

b. There are p=3 treatments. Therefore, there will be g=p(p-1)/2=3 pair wise comparisons. The experimentwise alpha=0.05, so the adjusted alpha level, alpha*=alpha/g=0.05/3=0.017. The critical Students' value for 15 degrees of freedom (df associated with MSE) and talpha/2~t0.017/2~t0.0083 is 2.70. From information, s=sqrt MSE=3.92.

Substitute the values in following formula and calculate Bij, critical difference.

B=t0.0083(s)sqrt(1/ni+1/nj)=(2.70*3.92)sqrt(1/6+1/60=6.11

Rank the sample means and place bar over pair of means that differ by less than 6.11. Any pair of means not connected by over bar implies a difference by corresponding population means.

17 19.33 24

48hrs 24hrs 12hrs

place bar over 17 and 19.33; 19.33 and 24. There is no bar placed over 17 and 24 hours. Thus conclude that there is significant difference in decision making for 12 hours and 48 hours sleep deprivaion conditions.

12 hours 24 hours 48 hours Xi Xi^2 Xi Xi^2 Xi Xi^2 22 484 18 324 16 256 19 361 21 441 20 400 34 1156 22 484 21 441 27 729 20 400 14 196 22 484 15 225 16 256 20 400 20 400 15 225 Sigma Xi=144 Sigma Xi^2=3614 Sigma Xi=116 Sigma Xi^2=2274 Sigma Xi=102 Sigma Xi^2=1774 Xk bar=24 Xk bar=19.33 Xk bar=17 Xbar=20.11(24+19.33+17)/3

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