A defense contractor has submitted to their customer, samples of a vehicle model for evaluation. The defense contractor was soon informed by their customer that most of the vehicles submitted were found to have many of the critical fasteners loosen as a result of some preliminary off-road testing.

Question

A defense contractor has submitted to their customer, samples of a vehicle model for evaluation. The defense contractor was soon informed by their customer that most of the vehicles submitted were found to have many of the critical fasteners loosen as a result of some preliminary off-road testing. A consultant is engaged and he suggests that submitting the prototype vehicles to whole body shaker tests will cause the initially defectively tightened critical fasteners to loosen an average of 4.5 N-m. This, the consultant claims, will minimize the “infant mortality” problem by identifying the incorrectly tightened fasteners prior to delivery to the customer. The consultant is authorized “to prove his allegation” and he subjects seven vehicles to whole body testing and he records the critical fastener torque levels before and after the whole body vibration test. He intends to perform a statistical analysis based on a “difference” approach.

The consultant intends to support his claim by comparing a 95% confidence interval for the mean “difference” of fastener torque.
Assume the difference of fastener torques to be approximately normally distributed.

Accepted Answer

1. Approach: Vehicles 1 2 3 4 5 6 7
Differences (D) −1.5 5.4 3.6 6.9 5.5 2.7 2.3
2. Data Yields: [latex]D_{AVG} = \overline{D} = 3.557[/latex]
[latex]S_{D} = 2.776[/latex]
3. [latex]∂= n −1 = 6 \alpha = 0.05;\longrightarrow \alpha / 2 = 0.025[/latex]
4. [latex]t_{\alpha/2,∂} = t_{0.025,6} = 2.447[/latex] (From Table 4.3)

5. [latex]\overline{D} -t_{\alpha /2,D}\frac{S_{D}}{\sqrt{n} }\lt \mu _{D} \lt \overline{D} +t_{\alpha /2,D}\frac{S_{D}}{\sqrt{n} }[/latex]

[latex]3.557-2.447\frac{2.776}{\sqrt{7} } \lt \mu _{D} \lt 3.557+2.447\frac{2.776}{\sqrt{7} }[/latex]

6. 0.990 < μ < 6.124

Since the estimated 4.5 Nm torque reduction (clamp load loss) falls within the calculated range, it can be asserted that the consultant’s claim is valid. That is, there will be a measureable loss of clamp load due to whole body shaker tests, which will permit identifying the improperly tightened critical fasteners.

	Vehicles Tested
	1	2	3	4	5	6	7
Critical-Fastener torque before shaker tests (Nm)	58.5	60.3	61.7	69	64	62.6	56.7
Critical-Fastener torque after shaker tests (Nm)	60	54.9	58.1	62.1	58.5	59.9	54.4

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

Question 4.DS.12: A defense contractor has submitted to their customer, sample......

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