Hypothesis Testing: Comparing Car Repair Costs with New vs Old Bumper Bars

In summary, a hire-car firm is considering a special bumper bar to reduce the cost of car repairs following minor accidents. The firm tested this bumper bar on a number of cars and recorded the cost of repairs over a one-year period. The summary statistics for the costs of repairs with the new and conventional bumper bars are given. To test if the new bumper bar reduces the mean cost of repairs by $500 or more, the null hypothesis is that there is no difference and the alternative hypothesis is that there is a difference. A one-tailed test would be used to determine the significance of this difference.
  • #1
nicholasch
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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...
 
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  • #2
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.

[tex]H_0: \mu_2 - \mu_1 \leq 500[/tex]
[tex]H_a: \mu_2 - \mu_1 > 500[/tex]

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

What statistic are you planning to use?
 

1. What is a hypothesis test?

A hypothesis test is a statistical method used to determine whether there is a significant difference between two or more groups or populations. It involves making an educated guess, or hypothesis, about the relationship between variables and then using data to determine the likelihood of the hypothesis being true.

2. How is a hypothesis test used in comparing car repair costs with new vs old bumper bars?

In this scenario, a hypothesis test would be used to determine if there is a significant difference in the average cost of car repairs when using new bumper bars compared to old bumper bars. The hypothesis would be that there is no difference in repair costs between the two types of bumper bars.

3. What is the null hypothesis?

The null hypothesis is the initial assumption that there is no significant difference between the two groups being compared. In this case, the null hypothesis would be that there is no difference in car repair costs between using new and old bumper bars.

4. How is the p-value used in a hypothesis test?

The p-value is a statistical measure that represents the probability of obtaining the observed results if the null hypothesis is true. In hypothesis testing, a p-value of less than 0.05 is typically considered significant, meaning that there is a less than 5% chance of obtaining the observed results if the null hypothesis is true. This would lead to rejecting the null hypothesis and accepting the alternative hypothesis.

5. Are there any limitations to using hypothesis testing in comparing car repair costs?

Yes, there are several limitations to consider when using hypothesis testing in this scenario. These include the sample size, the accuracy of the data, and the potential for other variables to impact the results. Additionally, hypothesis testing cannot prove causation, only correlation, so it is important to interpret the results carefully.

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