A special bumper was installed on selected vehicles in a large fleet. The dollar cost of body repairs was recorded for all vehicles that were involved in accidents over a 1-year period. Those with the special bumper are the test group and the other vehicles are the control group, shown below. Each "repair incident" is defined as an invoice (which might include more than one separate type of damage).

(a) Construct a 90 percent confidence interval for the true difference of the means assuming equal variances. Show all work clearly.
(b) Repeat, using the assumption of unequal variances with either Welch's formula for d.f. or the quick rule for degrees of freedom.
(c) Did the assumption about variances change the conclusion?
(d) Construct separate confidence intervals for each mean. Do they overlap?

Statistic Test Group Control Group Mean Damage Sample Std. Dev Repair Incidents x $1,101 s$696 n, 12 3 $1,766 -$838