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Question 4.DS.11: Dorf Motors, a vehicle manufacturer, is interested in perfor......

Dorf Motors, a vehicle manufacturer, is interested in performing a fastener integrity reliability study of their vehicles. Dorf Motors retains a consultant to compare their fastener recall data against Fota Motors fastener recall data as presented in the NHTSA website. Eight Dorf and five Fota vehicle platforms were chosen from 20 years of NHTSA recall data for analysis. A review of the data indicates that the eight Dorf vehicle types averaged 7 recalls per year with a standard deviation of 1.5 recalls per year and the five Fota vehicle platforms averaged 4 recalls per year with a standard deviation of 0.75 recalls per year.

The consultant assumes that the recalls per year for each vehicle type are approximately normally distributed with equal variances. He intends to construct a 90% confidence interval for the difference between the average recall per year for these two manufacturers’ vehicle types. He then sets forth his analysis as shown:

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1. n_{1} = 8 = n_{D}  n_{2} = 5 = n_{F} (small sample—use t distribution)
2. \overline{x}_{1} = 7  \overline{x}_{2} = 4\sigma_{1} = \sigma_{2} \longrightarrow but unknown
3. S_{1} = 1.5  S_{2} = 0.75
4. = 0.1
5. ∂ = n_{1} + n_{2} − 2   8 + 5 − 2 = 11

\begin{aligned} & \text { 6. } S_p^2=\frac{\left(n_1-1\right) S_1^2+\left(n_2-1\right) S_2^2}{n_1+n_2-2}=\frac{(8-1)(1.5)^2+(5-1)(0.75)^2}{8+5-2}=\frac{15.75+2.25}{11}=1.636 \\ & S_p=1.279 \\ & \text { 7. } t_{\alpha / 2, \partial}=t_{0.05,11}=1.796(\text { From Table 4.3) } \\ & \text { 8. }\left(\bar{x}_1-\bar{x}_2\right)-t_{\alpha / 2,∂} S_p\left[\frac{1}{n_1}+\frac{1}{n_2}\right]^{\frac{1}{2}}<\left(\mu_1-\mu_2\right)<\left(\bar{x}_1-\bar{x}_2\right)-t_{\alpha / 2,∂} S_P\left[\frac{1}{n_1}+\frac{1}{n_2}\right]^{\frac{1}{2}} \\ & \quad(7-4)-1.796(1.279)\left[\frac{1}{8}+\frac{1}{5}\right]^{\frac{1}{2}}<\left(\mu_1-\mu_2\right)<(7-4)+1.796(1.279)\left[\frac{1}{8}+\frac{1}{5}\right]^{\frac{1}{2}} \\ & \text { 9. } 1.691<\left(\mu_1-\mu_2\right)<4.309 \end{aligned}

Therefore, the average vehicle recall difference for the vehicle samples from the two manufacturers at a 90% confidence level will statistically range from 1.691 to 4.309.

TABLE 4.3
Percentile Values for Student’s t Distribution [5]
1 0.325 0.727 1.376 3.078 6.314 12.706 31.821 63.657
2 0.289 0.617 1.061 1.886 2.92 4.303 6.965 9.925
3 0.277 0.584 0.978 1.648 2.353 3.182 4.541 5.841
4 0.271 0.569 0.941 1.533 2.132 2.776 3.747 4.604
5 0.267 0.559 0.920 1.476 2.015 2.571 3.365 4.032
6 0.265 0.553 0.906 1.440 1.943 2.447 3.143 3.707
7 0.263 0.549 0.896 1.415 1.895 2.365 2.998 3.499
8 0.262 0.546 0.889 1.397 1.860 2.306 2.896 3.355
9 0.261 0.543 0.883 1.383 1.833 2.262 2.821 3.250
10 0.260 0.542 0.879 1.372 1.812 2.228 2.764 3.169
11 0.260 0.540 0.876 1.363 1.796 2.201 2.718 3.106
12 0.259 0.539 0.873 1.356 1.782 2.179 2.681 3.055
13 0.259 0.538 0.870 1.350 1.771 2.160 2.650 3.012
14 0.258 0.537 0.868 1.345 1.761 2.145 2.624 2.977
15 0.258 0.536 0.866 1.341 1.753 2.131 2.602 2.947
16 0.258 0.535 0.865 1.337 1.746 2.120 2.583 2.921
17 0.257 0.534 0.863 1.333 1.740 2.110 2.567 2.898
18 0.257 0.534 0.862 1.330 1.734 2.101 2.552 2.878
19 0.257 0.533 0.861 1.328 1.729 2.093 2.539 2.861
20 0.257 0.533 0.860 1.325 1.725 2.086 2.528 2.845
21 0.257 0.532 0.859 1.323 1.721 2.080 2.518 2.831
22 0.256 0.532 0.858 1.321 1.717 2.074 2.508 2.819
23 0.256 0.532 0.858 1.319 1.714 2.069 2.500 2.807
24 0.256 0.531 0.857 1.318 1.711 2.064 2.492 2.797
25 0.256 0.531 0.856 1.316 1.708 2.060 2.485 2.787
26 0.256 0.531 0.856 1.315 1.706 2.056 2.479 2.779
27 0.256 0.531 0.855 1.314 1.703 2.052 2.473 2.771
28 0.256 0.530 0.855 1.313 1.701 2.048 2.467 2.763
29 0.256 0.530 0.854 1.311 1.699 2.045 2.462 2.756
30 0.256 0.530 0.854 1.310 1.697 2.042 2.457 2.750
40 0.255 0.529 0.851 1.303 1.684 2.021 2.423 2.704
60 0.254 0.527 0.848 1.296 1.671 2.000 2.390 2.660
120 0.254 0.526 0.845 1.289 1.658 1.980 2.358 2.617
∞ 0.253 0.524 0.842 1.282 1.645 1.960 2.326 2.576

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