Consider the self rival data presented in Exercise 4.6.5. Recall that it is a paired design consisting of the pairs (Selfi, Rivali), for i = 1,..., 20, where Selfi and Rivali are the running times for...

Consider the self rival data presented in Exercise 4.6.5. Recall that it is a paired design consisting of the pairs (Selfi, Rivali), for i = 1,..., 20, where Selfi and Rivali are the running times for circling the bases for the respective treatments of Self motivation and Rival motivation. The data can be found in the file selfrival.rda. Let X_i
= Selfi − Rivali denote the paired differences and model these in the location model as X_i
= θ + e_i. Consider the hypotheses H₀
: θ = 0 versus H₁
: θ
0.

 (a) Obtain the signed-rank test statistic and p-value for these hypotheses. State the conclusion (in terms of the data) using the level 0.05.

 (b) Obtain the t test statistic and p-value and conclude using the level 0.05.

(c) To see the effect that an outlier has on these two analyses, change the 20th rival time from 17.88 to 178.8. Comment on how the analyses changed due to the outlier.

 (d) Obtain 95% confidence intervals for θ for both analyses for the original data and the changed data. Comment on how the confidence intervals changed due to the outlier.

Exercise 4.6.5

On page 373 Rasmussen (1992) discussed a paired design. A baseball coach paired 20 members of his team by their speed; i.e., each member of the pair has about the same speed. Then for each pair, he randomly chose one member of the pair and told him that if could beat his best time in circling the bases he would give him an award (call this response the time of the “self” member). For the other member of the pair the coach’s instruction was an award if he could beat the time of the other member of the pair (call this response the time of the “rival” member). Each member of the pair knew who his rival was. The data are given below, but are also in the file selfrival.rda. Let μd be the true difference in times (rival minus self) for a pair. The hypotheses of interest are H₀
: μ_d
= 0 versus H₁
: μ_d
<>

self: 16.20 16.78 17.38 17.59 17.37 17.49 18.18 18.16 18.36 18.53

(a) Obtain comparison boxplots of the data. Comment on the comparison plots. Are there any outliers?

 (b) Compute the paired t-test and obtain the p-value. Are the data significant at the 5% level of significance?

 (c) Obtain a point estimate of μ_d
and a 95% confidence interval for it.

 (d) Conclude in terms of the problem