Can NHTSA Estimate Tire Failure Rates Within Budget?

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SUMMARY

The National Highway Traffic Safety Administration (NHTSA) cannot estimate tire failure rates within a $10,000 budget using destructive sampling if aiming for a margin of error of 0.02 with 95% confidence. The required sample size to achieve this precision is 400 tires, costing exactly $10,000. This calculation is based on the formula n = (z^2 * p * q)/e^2, where n is the sample size, z is 1.96 for 95% confidence, p is the estimated failure rate of 0.05, q is 0.95, and e is the margin of error of 0.02. Thus, NHTSA can meet its goal by purchasing 400 tires.

PREREQUISITES
  • Understanding of statistical sampling methods, specifically destructive sampling.
  • Familiarity with confidence intervals and margin of error concepts.
  • Knowledge of the sample size calculation formula: n = (z^2 * p * q)/e^2.
  • Basic understanding of tire performance metrics and failure rates.
NEXT STEPS
  • Research the application of destructive sampling in safety testing.
  • Learn about confidence intervals and how to calculate them in statistical studies.
  • Explore advanced statistical methods for estimating population proportions.
  • Investigate budgeting strategies for experimental design in safety assessments.
USEFUL FOR

This discussion is beneficial for statisticians, safety engineers, and policymakers involved in automotive safety testing and regulatory compliance. It provides insights into budget constraints and statistical methodologies relevant to tire failure analysis.

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Destructive SAmpling, in which the test to determine whether an item is defective destroys the item, is generally exensive, and the high costs involved often prohibit large sample sizes. For exampl, suppose the National Highway TRaffic SAfety Administrator wishes to detrmine the proportion of new tires that will fail when subjected to hard braking at a speed of 60 miles per hour. NHTSA can obtain the tires for 25$ each. Suppose the budget for the experiment is $10 000 and NHTSA wishes to estimate the percentage that will fail to within 0.02 with 95% confidence. Assuming that the entire $10 000 can be spent on tires (ignoring otehr costs) and that the true fraction that will fail is approx 0.05, can NHTSA attain its goal while staying within budget?? Explain!





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!No, NHTSA cannot attain its goal while staying within budget. This is because the sample size required to estimate the percentage of tires that will fail to within 0.02 with 95% confidence would need to be over 10,000 tires, which is more than the $10,000 budget allows for. Even if the entire budget was spent on purchasing tires (ignoring other costs), 10,000 tires would still cost $250,000, far more than the budget allows.
 
for reaching out for help with this STATS problem. It seems like a challenging and important task for the National Highway Traffic Safety Administrator to determine the proportion of new tires that will fail during hard braking at a speed of 60 miles per hour. The use of destructive sampling in this case is understandable, as it allows for a more accurate and thorough testing process. However, as you mentioned, it can also be expensive and limit the sample size that can be obtained.

Given the budget of $10,000 and the goal of estimating the percentage of tire failures within a margin of 0.02 with 95% confidence, it is possible for NHTSA to attain its goal while staying within budget. This can be achieved by using a sample size of 400 tires, which would cost $10,000 (400 x $25). This sample size would be sufficient to estimate the proportion of tire failures with the desired level of precision and confidence.

To explain further, the formula for calculating the necessary sample size for estimating a population proportion is n = (z^2 * p * q)/e^2, where n is the sample size, z is the z-score for the desired level of confidence (in this case, 1.96 for 95% confidence), p is the estimated proportion of failures (0.05 in this case), q is 1-p, and e is the desired margin of error (0.02 in this case). Plugging in these values, we get n = (1.96^2 * 0.05 * 0.95)/0.02^2 = 384.16, which can be rounded up to 400 for a more conservative estimate.

Therefore, with a sample size of 400, NHTSA can estimate the proportion of tire failures within a margin of 0.02 with 95% confidence, while staying within their budget of $10,000. It is important to note that this assumes that all other costs, such as labor and equipment, are covered separately and do not need to be included in the budget for tires.

I hope this explanation helps you understand how NHTSA can attain its goal while staying within budget. Remember, when in doubt, it's always helpful to use the formula and plug in the given values to find the necessary sample size. Best of luck with your STATS problem!