MEMORANDUM TO: Honorable Farad Hadi FROM: Etienne Labastie, Engineering Expert Witness DATE: April 18, 2019 RE: Tolford V. The Automotive Shaft Company Question Presented Are the engineers claims of the manufactured shafts within acceptable limits in terms of wear valid? Brief Answer I believe that the recent deaths from the catastrophic engine failures was not due to faulty bearings in the manufactured shafts and that the engineer’s claims are valid. Facts Throughout this trial, lawyers have sued the Automotive Shaft Company about their manufactured shafts exceeding shaft wear parameters, resulting in recent deaths from catastrophic vehicle failures. However, the company’s engineers claim that their manufactured shafts are within acceptable limits in terms of wear. In order to answer this question, I was provided with variable such as the standard deviation (S), the mean of the data (), how many combustion engines were tested (n), and the normal distribution mean (). The variables given were standard deviation = 1.25, the mean of the data = 3.72, the number of combustion engines tested = 8, and the normal distribution mean = 3.5. Discussion In the case of whether the recent deaths from catastrophic engine failures was due to faulty bearings in the manufactured shafts, I have determined that the engineers claim of their manufactured shafts meeting parameters is valid. In order to come to this conclusion, engineers presented me with data such as the number of combustion engines tested, the standard deviation, mean of the data, and the normal distribution mean. With this information, I was able to complete a ttest/hypothesis test in order to determine whether the wear of the shaft should be eliminated as a suspect. I first calculated the test statistic value using these variables provided to me. After calculating this value, I pulled up a t-curve tail area table in order to observe the p-value for this calculated t-value. This approach is best to use because it helps calculate the t-value distribution and determine whether a null hypothesis is true or can be rejected. In the case of this trial, the null hypothesis is on the wear of the manufactured shafts from the company fitting parameters. After performing the calculations, the test statistic value was calculated to be t=0.498. I then looked up the critical value for this t-value in the table and observed a value of 1.895. Then, using this data, I looked at the t curve tail areas table at t=0.498 for 7 df, and observed that the p-value was equal to 0.316. After calculating the hypothesis test value, it was determined that the null hypothesis, H0, cannot be rejected from the trial, meaning that the company’s engineers claims appear to be valid about their manufactured shafts have a shaft wear within acceptable limits. Counterargument One counterargument that was presented in this trial was that the number of combustion engines tested upon was a small sample, meaning that the t-value may have been lower, resulting in a higher probability that the shaft wear exceeded limits and the engineers claims not being valid. However, since this was a small sample of combustion engines tested on, then the t-score will increase, resulting in a lower probability that the engineers claim will not be valid. Conclusion In conclusion, the hypothesis test was completed, and it was calculated that the test statistic value, t-value, was calculated to be 0.498. Then, the p-value, probability, was calculated to be 0.316, which supports that the null hypothesis, H0, cannot be rejected. One of the counterarguments that was presented in this trial was that the number of combustion engines tested was small. However, if the number of combustion engines tested were increased, then the evidence supporting the engineers claim would be supported since this would result in the increase of the t-value and decrease the probability that the shaft wear exceeded the limit. In conclusion, the company’s engineers claim that their manufactured shafts have a shaft wear within acceptable limits is valid. Addendum 1: Statistical Calculations Mathematical Analysis Figure 1: Equation Used to Calculate the Test Statistic Value t=(x−μ)Sn • • • t = Test statistic value = 0.498 x = Mean of Data = 3.72 n = Number of Combustion Engines Tested = 8 • • μ = Normal Distribution Mean = 3.5 S = Standard Deviation = 1.25 Figure 2: Calculation of the Test Statistic Value t=(3.72−3.50)1.258=0.498 Figure 3: Calculation of the P-Value for the Calculated Test Statistic Value in Excel Memorandum To: The Honorable Farah Hadi CC: Joe Schmoe, Clerk of the Court From: Trevor Patrick, Engineering Expert Witness Date: April 19, 2019 RE: Tolford’s claim that The Automotive Shaft Company’s negligence has led to deaths Question Presented Has an excess amount of wear in shafts produced by The Automotive Shaft Company been the cause of catastrophic failures in extreme weather conditions, leading to deaths? Brief Answer It is unlikely that excess shaft wear has led to deaths in extreme weather conditions. While it is not possible to come to a firm conclusion based on only a small amount of data, statistical analysis on the supplied data shows a low probability that excess shaft wear has led to deaths. Facts Shaft wear in excess of 3.50(ten thousandths of an inch) could lead to catastrophic failures in extreme weather conditions. Tolford had filed a class action legal suit against The Automotive Shaft Company, claiming that these shafts have caused recent deaths of owners of vehicles that have these shafts. Discussion Can excess wear of shafts produced by The Automotive Shaft Company be blamed for multiple fatalities? After statistical analysis, it is concluded that it is unlikely that excess shaft wear has led to deaths in extreme weather conditions. The best way to approach this question from a statistical standpoint is to consider the probability of two cases: when the average shaft wear is 3.50(10-4 inch), and when the average shaft wear is greater than 3.50(10-4 inch). In a situation like this one, the data naturally forms a bell curve (normal distribution). This enables accurate conclusions to be drawn from a small sample of data, given some initial data about the distribution (mean and standard deviation). See Addendum 1 for an in-depth analysis. The analysis indicates strong evidence that supports The Automotive Shaft Company’s claim that shaft wear is within acceptable limits. While Tolford claims that such a small sample size of 8 engines isn’t enough to draw meaningful conclusions, the nature of the data (natural bell curve) allows such conclusions to be formed, as explained above. Tolford’s claim that The Automotive Shaft Company’s negligence has led to deaths 1 Conclusion On these facts, the court will probably find that The Automotive Shaft Company should not be held responsible for the recent deaths. While the data is indeed a small sample size, the conclusion is still viable. To further substantiate the evidence, and perhaps provide an even more clear conclusion, would require a larger sample size. An additional investigation to find out if another part of the car could have been the issue would be a good course of action. Tolford’s claim that The Automotive Shaft Company’s negligence has led to deaths 2 Addendum 1: Statistical Calculations Notation and Given Information: X = shaft wear, (10 -4 inch) X is distributed normally: x ~ N(μ = mean, σ2 = variance) Sample standard deviation (s) = 1.25 Two hypotheses are H0 : μ = 3.50, and Hα : μ > 3.50 Significance level (α) = 0.05 Number of samples (n) = 8 Average amount of shaft wear (X) = 3.72 Calculations t= 𝑋−𝜇0 s/√𝑛 = 3.72−3.50 1.25/√8 = 0.498 R : t > tα,n-1 = t0.05,7 = 1.895 T = 0.498 ∉ R =[1.895, infinity] Conclusion By looking at the 0.5 row and the 7 column in a t-curve tail areas table, we see that the corresponding area to the right of t is 0.316 (p-value). This is the probability that the average shaft wear of all the shafts made by the company is not 3.50 (10-4 inches). We can confidently say this because our t = 0.498 is close to 0.5, therefore the p-value ≈ 0.316. This is a low probability, concluding that most shafts wear roughly 3.50 (10-4 inches). Note: A small p-value (≤ 0.05) indicates strong evidence against the null hypothesis, thus rejecting the null hypothesis. The shaded region represents the probability of a more extreme result, assuming the null hypothesis is true. This is a low probability that an extreme result (shaft wear greater than 3.50x10-4 inches) will occur. Tolford’s claim that The Automotive Shaft Company’s negligence has led to deaths 3 Memorandum To: Honorable Farah Hadi CC: The Jury From: Justin Osagie, Engineering Expert Witness Date: April 19, 2019 RE: Mr. Tolford’s claim of failures in shaft wear from the Automotive Shaft Company Question Presented Florida Second District Court. Is the Automotive Shaft Company at fault for the catastrophic failures in extreme weather conditions of shafts they produce? Brief Answer No, the Automotive Shaft Company is not at fault for the catastrophic failures in extreme weather conditions of the shafts they produce. Facts The Automotive Shaft Company produces the shafts that are brought to court in this class action lawsuit. In testing eight internal combustion engines having a copper lead as a bearing material, the average shaft wear was 3.72. Shaft wear in excess of 3.50 could lead to catastrophic failures in extreme weather conditions. Engineers of the Automotive Shaft Company believe the catastrophic vehicle failures for engines with this shaft are due to faulty bearings. Discussion There is not enough evidence to justify that the true average shaft wear exceeds 3.5. This conclusion came from using the one-sided t test since there were two hypothesis. The Null Hypothesis, which the normal shaft wear, was 3.5, and the Alternative Hypothesis in which the normal shaft wear was greater than 3.5. With the significance level at 0.05, the result from the t test (0.4978) found that there was not enough evidence to prove that shaft wear was at fault of the Automotive Shaft Company. The one sided t Test was the best approach to this problem since there were only two hypothesis, the null (= 3.5) and the alternative hypothesis (>3.5). The outcome of the analysis was a final P-value of 0.316 which is greater than 0.05. This leads to not enough significant evidence to prove that the null hypothesis, the normal shaft wear being 3.5, was incorrect. This outcome leads to my answer that the Automotive Shaft Industry is not at fault for the catastrophic failures in extreme weather conditions for the shafts they produce. The Automotive Shaft Company’s claim to not be at fault is correct while the reasoning for the vehicle failures cannot be determined with the provided data. Tolford v. Automotive Shaft Company 1 Conclusion Since finding that the Automotive Shaft Company is not at fault the outcome of the case should find the company not at fault for the vehicle failures. Addendum: Statistical Calculations Tolford v. Automotive Shaft Company 2 The One-Sample t Test: t = (x-bar – population mean) / (s / sqrt (n)) x-bar = 3.72 s = 1.25 test level .05 H0 : μ = 3.50 (There is no significant shaft wear) H α: μ > 3.50 (There is significant shaft wear greater than 3.5) n=8 (3.72 – 3.5) / (1.25 / sqrt (8)) t = .4978 8 – 1 = 7 df P-value = 0.316 Tolford v. Automotive Shaft Company 3 Case: Tolford v. The Automotive Shaft Company Addresse: The Honorable Farah Hadi Hello, I have been asked to analyze the wear of shafts on internal combustion engine cars in this court case. According to my calculations, the shafts are the cause of these fatal accidents that have been happening. There is a set statistical test that one can perform that gives an accurate interval of the mean of a sample. I will explain in detail each step of the process so the jurors can follow along. To begin the experiment, I started by assuming my first hypothesis was equal to the fatal value of 3.5 inches in diameter. If this hypothesis cannot be rejected, it is assumed that the shafts are failing. I decided the alternative hypothesis in the experiment were to be that the diameter of the shaft was greater than the failure value of 3.5 inches. Fortunately, the standard deviation and sample size were also given with the information, with a standard deviation of 1.25 and a sample size of 8. The alpha value for the experiment was also determined to be .05, or 5 percent. Using z tables and the formulas to calculate the z scores of the sample, I came to the conclusion that the shafts were the cause of these accidents, as I could not disprove the original hypothesis, therefore the hypothesis is true.
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