Testing if new bumper bar reduces repair costs by more than $500

[nicholasch]

Homework Statement

A hire-car firm is considering a special bumper bar designed to lower the cost of car repairs following minor accidents. This bumper bar is more expensive than the conventional one. It will only be introduced permanently if any reduction in the cost of repairs is judged to be sufficiently large. The firm had the special bumper bar installed on a number of its cars. Over the following one-year period, the cost of car repairs was recorded for all cars involved in minor accidents. The summary statistics for these costs for cars with the new bumper bar, and with the conventional bumper bar, are given in the following table.

Statistics:
New Bumper Bar - Mean cost of repair (xbar1=$1101), Standard Dev of cost (s1=$696), number of repair incidents (n1=12)
Old Bumper Bar - Mean cost of repair (xbar2=$1766), Standard Dev of cost (s2=$838), number of repair incidents (n2=9)

(a) The new bumper bar costs an extra $500. Test whether the reduction in the mean cost of repairs is greater than $500. [Use a significance level of a =0.10 and assume the population variances for the costs of repairs for the new and old bumper bars are the same.]


The attempt at a solution
So i reckon that H0:x1-x2>500 and H1:x1-x2<500, I am not really sure where to proceed though...

 

[Mark44]

Your null hypothesis should be that the new bumper bar makes no difference, and the alternate hypothess is that it does make a difference. In terms of your problem, I believe these are what your hypotheses should be.

$$H_0: \mu_2 - \mu_1 \leq 500$$
$$H_a: \mu_2 - \mu_1 > 500$$

With these hypotheses, you would want a one-tailed test.

What statistic are you planning to use?