Problem Set

Problem No 1.

A fatigue life of aluminum bolt is tested by subjecting it to a cyclic bending stress and counting the number of cycles until failure. The following data on the number of cycles until failure was taken over a series of test bolts:

23	55	123	121	248
56	321	69	287	96
124	235	86	156	214
489	112	158	145	187
44	235	470	42	234
256	89	73	86	156
77	355	55	155	422
452	423	149	423	123
123	199	163	189	186
14	32	185	354	145
55	45	166	185	132
89	86	72	69	52
12	256	135	75	42
125	175	255	355	468
256	123	230	369	358
243	108	175	286	210
125	149	136	186	165
189	136	156	178	45
175	186	289	136	136
234	175	236	259	196
203	145	244	145	324

Using intervals 0-100, 100-200, etc. construct a relative frequency distribution and histogram for the fatigue life of the bolt and test stress level.
If the bolt is required to withstand at least 100 cycles at this stress level, what proportion of the bolts would be considered satisfactory?

Problem No 2.

Americium 241 (²⁴¹Am) is a radioactive material used in the manufacture of smoke detectors. The article "Retention and Dosimetry of Injected ²⁴¹Am in Beagles" (Radiation Research, 1984, pp. 564-575) described a study in which 55 beagles were injected with a dose of ²⁴¹Am (proportional to animal's weights). Skeletal retention of ²⁴¹Am (mCi/kg) was recorded for each beagle, resulting in the given data:

0.196	0.451	0.498	0.411	0.324
0.190	0.489	0.300	0.346	0.448
0.188	0.399	0.305	0.304	0.287
0.243	0.334	0.299	0.292	0.419
0.236	0.315	0.447	0.585	0.291
0.186	0.375	0.419	0.335	0.332
0.292	0.408	0.349	0.324	0.301
0.333	0.441	0.399	0.303	0.318
0.468	0.345	0.306	0.367	0.345
0.428	0.320	0.412	0.337	0.353
0.357	0.393	0.354	0.361	0.329

Construct a relative frequency distribution and histogram of this data based on 10 intervals starting from 0.1 and having width 0.05 each. Find mean, median and standard deviation.
Using a cumulative frequency plot, estimate 10th, 50th, and 90th percentile values.

23	55	123	121	248
56	321	69	287	96
124	235	86	156	214
489	112	158	145	187
44	235	470	42	234
256	89	73	86	156
77	355	55	155	422
452	423	149	423	123
123	199	163	189	186
14	32	185	354	145
55	45	166	185	132
89	86	72	69	52
12	256	135	75	42
125	175	255	355	468
256	123	230	369	358
243	108	175	286	210
125	149	136	186	165
189	136	156	178	45
175	186	289	136	136
234	175	236	259	196
203	145	244	145	324

23	55	123	121	248
56	321	69	287	96
124	235	86	156	214
489	112	158	145	187
44	235	470	42	234
256	89	73	86	156
77	355	55	155	422
452	423	149	423	123
123	199	163	189	186
14	32	185	354	145
55	45	166	185	132
89	86	72	69	52
12	256	135	75	42
125	175	255	355	468
256	123	230	369	358
243	108	175	286	210
125	149	136	186	165
189	136	156	178	45
175	186	289	136	136
234	175	236	259	196
203	145	244	145	324

23	55	123	121	248
56	321	69	287	96
124	235	86	156	214
489	112	158	145	187
44	235	470	42	234
256	89	73	86	156
77	355	55	155	422
452	423	149	423	123
123	199	163	189	186
14	32	185	354	145
55	45	166	185	132
89	86	72	69	52
12	256	135	75	42
125	175	255	355	468
256	123	230	369	358
243	108	175	286	210
125	149	136	186	165
189	136	156	178	45
175	186	289	136	136
234	175	236	259	196
203	145	244	145	324