# Acceptance Sampling Plans from Life Tests Based on Percentiles of Half Normal Distribution

Authors: B. Srinivasa Rao; Ch. Srinivasa Kumar; K. Rosaiah
Journal: Journal of Quality and Reliability Engineering
 (2013)

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## Abstract

The design of acceptance sampling plans is developed under truncated life testing based on the percentiles of half normal distribution. The minimum sample size necessary to ensure the specified life
percentile is obtained under a given consumer’s risk. The operating characteristic values of the sampling plans as well as the producer’s risk are presented. The results are illustrated by examples.


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## Body

## 1. Introduction

Acceptance sampling is concerned with inspection and decision making regarding lots of products and constitutes one of the oldest techniques in quality control. If the lifetime of the product represents the quality characteristics of interest, the acceptance sampling is as follows: a company receives a shipment of product from a vendor. This product is often a component or raw material used in the company’s manufacturing process. A sample is taken from the lot and the relevant quality characteristic of the units in the sample is inspected. On the basis of the information in the sample, a decision is made regarding lot disposition. Traditionally, when the life test indicates that the mean life of products exceeds the specified one, the lot of products is accepted, otherwise it is rejected. Accepted lots are put into production, while rejected lots may be returned to the vendor or may be subjected to some other lot disposition actions. For the purpose of reducing the test time and cost, a truncated life test may be conducted to determine the sample size to ensure a certain mean life of products when the life test is terminated at a timet0, and the number of failures does not exceed a given acceptance number c.A common practice in life testing is to terminate the life test by a predetermined timet0 and note the number of failures. One of the objectives of these experiments is to set a lower confidence limit on the mean life. It is then to establish a specified mean life with a given probability of at least p* which provides protection to consumers. The test may be terminated before the time is reached or when the number of failures exceeds the acceptance number c in which case the decision is to reject the lot.Studies regarding truncated life tests can be found in Epstein [1], Sobel and Tischendrof [2], Goode and Kao [3], Gupta and Groll [4], Gupta [5], Fertig and Mann [6], Kantam and Rosaiah [7], Baklizi [8], Wu and Tsai [9], Rosaiah and Kantam [10], Rosaiah et al. [11], Tsai and Wu [12], Balakrishnan et al. [13], Srinivasa Rao et al. [14], Srinivasa Rao et al. [15], Aslam et al. [16], and Srinivasa Rao et al. [17]. All these authors designed acceptance sampling plans based on the mean life time under a truncated life test. In contrast, Lio et al. [18] considered acceptance sampling plans for percentiles using truncated life tests and assuming Birnbaum-Saunders distribution. Srinivasa Rao and Kantam [19] developed similar plans for the percentiles of log-logistic distribution.Normal distribution is the most preferred distribution in statistical studies. But for life test models it is not suitable distribution because of its domain[-∞,+∞]. Half normal distribution is an increasing failure rate (IFR) model which is most useful in reliability studies. Because of this IFR nature with domain [0,+∞] we are motivated to study this distribution.In this paper, acceptance sampling plans are developed for percentiles of half normal distribution life test and are given in Section2. Operating characteristic (OC) and producer’s risk are given in Section 3. Examples based on real fatigue life data sets are provided for an illustration in Section 4. The paper is closed with summary and conclusions in Section 5.


## 2. Acceptance Sampling Plans

The probability density function (pdf) of a half normal distribution is given by(1)p(x)=2θπe-x2θ2/π,x≥0.Its cumulative distribution function (cdf) is(2)P(x)=erf(θxπ),x≥0,
where erf is an error function and θ is a parameter. Assume that the life time of a product follows half normal distribution with σ as scale parameter. Its cumulative distribution function F(·) is given by
(3)F(t)=erf(θ(t/σ)π),t≥0,σ>0.Given0<q<1, the 100th percentile is given by
(4)tq=σπθerf-1(q).Substitutingσ in (3) in the scaled form we get
(5)F(t)=erf(θ(t/tq)(π/θ)erf-1(q)π)⟹erf(δ0erf-1(q)),
where δ0=t/tq. Here erf-1 is simulated by
(6)erf-1(z)=12π[z+π12z3+7π2480z5+127π340320z7+4369π45806080z9⋯].
A common practice in life testing is to terminate the life test by a predetermined time t, to require the probability of rejecting a bad lot to be at least p*, and to have the maximum permissible number of bad items to accept the lot equal to c. The acceptance sampling plan for percentiles under a truncated life test is to set up the minimum sample size n for a given acceptance number c such that the consumer’s risk, the probability of accepting a bad lot, does not exceed 1-p*. A bad lot means that the true 100th percentile, tq, is below a specified percentile, tq0. Thus the probability p* is a confidence level in the sense that the chance of rejecting a bad lot with tq<tq0 is at least equal to p*. Therefore, for given p*, the proposed acceptance sampling plan can be characterized by the triplet (n,c,δ0)=(n,c,t/tq0), where δ=t/tq0.We consider large sized lots so that the binomial distribution can be applied. The problem is to determine for given values ofp*, tq0, and c the smallest positive integer n required to assert that tq>tq0 must satisfy the relation
(7)∑i=0c‍p0i(1-p0)n-i≤1-p*,
where p0=Ft(δ0) is the probability of a failure during the time t=δtq0 for the specified 100qth percentile of lifetime tq0. The value p0 depends only on δ0=t/tq0. Since ∂Ft(δ)/∂δ>0, it is an increasing function of δ. Accordingly, we have
(8)Ft(δ)≤Ft(δ0)⟺δ≤δ0
or equivalently
(9)F(t;tq)<F(t;tq0)⟺tq≥tq0.
The smallest sample size n satisfying (9) can be obtained for any given q, t/tq0, or  p*. In contrast, only the input values t/σ, p* are needed to calculate the smallest sample size n. In particular, if we take q=0.50, the value of n would become “the smallest sample size required to test that the population median life exceeds a given specified value.” Half normal distribution is a skewed distribution; for our present skewed population the median is a more appropriate average for decision making about the quality of the life than population mean. Thus we may conclude that population median based sampling plans of half normal distribution model are more economical than those based on population mean with respect to sample size. To save space, only the results of small sample sizes for q=0.50; p*=0.75,0.90,0.95,0.99;c=0(1)10; δ=0.1,0.2,0.4,0.6,0.8,1.0,1.5,2.0,2.5,3.0 are displayed in Table 1.Table 1
Minimum sample sizes necessary to assert the median life of a product.
p
*
c
t
/
t
0.5
0
0.1
0.2
0.4
0.6
0.8
1.0
1.5
2.0
2.5
3.0
0.75
0
26
13
6
4
3
3
2
2
2
1
0.75
1
50
25
12
8
6
5
3
3
2
2
0.75
2
72
36
18
12
9
7
5
4
4
3
0.75
3
94
47
23
16
12
10
7
5
5
4
0.75
4
116
58
29
19
15
12
8
7
6
5
0.75
5
137
68
34
23
17
14
10
8
7
6
0.75
6
158
79
39
26
20
16
11
9
8
8
0.75
7
179
89
45
30
23
18
13
11
9
9
0.75
8
200
100
50
33
25
21
14
12
10
10
0.75
9
220
110
55
37
28
23
16
13
12
11
0.75
10
241
120
60
40
31
25
18
14
13
12
0.90
0
42
21
10
7
5
4
2
2
2
2
0.90
1
71
35
17
11
8
7
4
3
3
2
0.90
2
98
48
24
16
12
9
6
5
4
4
0.90
3
123
61
30
20
15
12
8
6
5
5
0.90
4
147
73
36
24
18
14
10
8
6
6
0.90
5
171
85
42
28
21
17
11
9
8
7
0.90
6
194
96
48
32
24
19
13
10
9
8
0.90
7
217
108
53
35
27
21
15
12
10
9
0.90
8
240
119
59
39
29
24
16
13
11
10
0.90
9
262
130
65
43
32
26
18
14
12
11
0.90
10
284
141
70
47
35
28
20
16
14
12
0.95
0
55
27
13
8
6
5
3
2
2
2
0.95
1
87
43
21
14
10
8
5
4
3
3
0.95
2
115
57
28
18
13
11
7
5
4
4
0.95
3
142
70
34
23
17
13
9
7
6
5
0.95
4
168
83
41
27
20
16
11
8
7
6
0.95
5
193
96
47
31
23
18
12
10
8
7
0.95
6
218
108
53
35
26
21
14
11
9
8
0.95
7
242
120
59
39
29
23
16
12
11
10
0.95
8
266
132
65
43
32
26
18
14
12
11
0.95
9
289
143
71
47
35
28
19
15
13
12
0.95
10
312
155
77
51
38
30
21
16
14
13
0.99
0
84
41
20
13
9
7
4
3
2
2
0.99
1
121
59
29
19
14
11
7
5
4
3
0.99
2
154
76
37
24
17
14
9
7
5
5
0.99
3
184
91
44
29
21
17
11
8
7
6
0.99
4
212
105
51
33
25
19
13
10
8
7
0.99
5
240
119
58
38
28
22
15
11
9
8
0.99
6
267
132
65
42
31
25
16
13
10
9
0.99
7
293
145
71
47
35
27
18
14
12
10
0.99
8
319
158
77
51
38
30
20
15
13
12
0.99
9
345
171
84
55
41
33
22
17
14
13
0.99
10
370
183
90
59
44
35
24
18
15
14


## 3. Operating Characteristic of the Sampling Plan and Producer’s Risk

The operating characteristic (OC) functionL of the sampling plan (n,c,t/tq0) is the probability of accepting a lot as a function of p=Ft(δ) with δ0=t/tq. It is given as
(10)L(p)=∑i=0c‍(ni)pi(1-p)n-i.
Therefore, we have
(11)p=Ft(δ)=F(ttq01dq),
where dq=tq/tq0. Using (10) the OC values can be obtained for any sampling plan (n,c,t/t0.50) (Table 5). To save space we present the OC values for sampling plans with q=0.50;p*=0.75,0.90,0.95,0.99; δ=0.1,0.2,0.4,0.6,0.8,1.0,1.5,2.0,2.5,3.0; t0.5/t0.50=1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5,5.0; for c=1 and c=5  which are given in Table 2 and Table 3, respectively.Table 2
OC values of sampling plans.
c
=
1
t
/
t
0.5
0
p
*
n
δ
0.01
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.75
50
0.1
0.2420
0.4605
0.6093
0.7075
0.7741
0.8207
0.8545
0.8796
0.8989
25
0.2
0.2345
0.4576
0.6090
0.7084
0.7754
0.8221
0.8559
0.8810
0.9001
12
0.4
0.2408
0.4764
0.6301
0.7281
0.7928
0.8373
0.8690
0.8924
0.9101
8
0.6
0.2283
0.4759
0.6347
0.7342
0.7989
0.8430
0.8742
0.8970
0.9142
6
0.8
0.2173
0.4788
0.6425
0.7427
0.8070
0.8502
0.8805
0.9026
0.9191
5
1.0
0.1875
0.4609
0.6328
0.7373
0.8038
0.8482
0.8793
0.9018
0.9185
3
1.5
0.2309
0.5616
0.7262
0.8143
0.8662
0.8992
0.9214
0.9370
0.9484
3
2.0
0.0832
0.4276
0.6317
0.7464
0.8157
0.8602
0.8905
0.9120
0.9272
2
2.5
0.1752
0.6334
0.7938
0.8680
0.9084
0.9327
0.9484
0.9593
0.9670
2
3.0
0.0843
0.5920
0.7711
0.8535
0.8983
0.9253
0.9428
0.9548
0.9634
0.90
71
0.1
0.0993
0.2723
0.4274
0.5466
0.6354
0.7019
0.7524
0.7915
0.8221
35
0.2
0.0980
0.2752
0.4332
0.5534
0.6424
0.7085
0.7586
0.7971
0.8272
17
0.4
0.0961
0.2833
0.4473
0.5695
0.6583
0.7235
0.7723
0.8095
0.8384
11
0.6
0.0953
0.2950
0.4654
0.5891
0.6772
0.7409
0.7879
0.8235
0.8509
8
0.8
0.0958
0.3109
0.4880
0.6124
0.6989
0.7604
0.8053
0.8388
0.8645
7
1.0
0.0625
0.2634
0.4449
0.5767
0.6698
0.7365
0.7854
0.8221
0.8503
4
1.5
0.0928
0.3766
0.5734
0.6949
0.7725
0.8243
0.8605
0.8866
0.9061
3
2.0
0.0832
0.4276
0.6317
0.7464
0.8157
0.8602
0.8905
0.9120
0.9277
3
2.5
0.0237
0.3441
0.5687
0.7000
0.7805
0.8329
0.8688
0.8942
0.9130
2
3.0
0.0843
0.5930
0.7711
0.8535
0.8983
0.9253
0.9428
0.9548
0.9364
0.95
87
0.1
0.0484
0.1766
0.3176
0.4390
0.5362
0.6127
0.6730
0.7209
0.7594
43
0.2
0.0468
0.1773
0.3210
0.4440
0.5420
0.6186
0.6788
0.7264
0.7645
21
0.4
0.0441
0.1804
0.3302
0.4566
0.5558
0.6325
0.6920
0.7388
0.7760
14
0.6
0.0377
0.1753
0.3298
0.4595
0.5605
0.6380
0.6978
0.7445
0.7815
10
0.8
0.0404
0.1948
0.3602
0.4932
0.5935
0.6688
0.7259
0.7699
0.8043
8
1.0
0.0352
0.1951
0.3671
0.5033
0.6047
0.6798
0.7363
0.7795
0.8131
5
1.5
0.0354
0.2431
0.4397
0.5795
0.6761
0.7441
0.7933
0.8297
0.8575
4
2.0
0.0194
0.2436
0.4558
0.6002
0.6969
0.7633
0.8105
0.8450
0.8710
3
2.5
0.0237
0.3441
0.5687
0.7000
0.7805
0.8329
0.8688
0.8942
0.9130
3
3.0
0.0054
0.2983
0.5323
0.6726
0.7597
0.8166
0.8557
0.8836
0.9041
0.99
121
0.1
0.0098
0.0662
0.1603
0.2633
0.3593
0.4433
0.5148
0.5749
0.6256
59
0.2
0.0100
0.0695
0.1679
0.2740
0.3718
0.4565
0.5280
0.5878
0.6378
29
0.4
0.0086
0.0688
0.1709
0.2807
0.3811
0.4672
0.5392
0.5991
0.6489
19
0.6
0.0075
0.0694
0.1766
0.2908
0.3938
0.4812
0.5536
0.6133
0.6626
14
0.8
0.0066
0.0714
0.1851
0.3044
0.4103
0.4987
0.5712
0.6303
0.6788
11
1.0
0.0059
0.0751
0.1971
0.3221
0.4307
0.5199
0.5919
0.6501
0.6974
7
1.5
0.0047
0.0942
0.2439
0.3841
0.4975
0.5859
0.6446
0.7083
0.7509
5
2.0
0.0042
0.1328
0.3177
0.4694
0.5819
0.6644
0.7257
0.7721
0.8079
4
2.5
0.0029
0.1730
0.3843
0.5395
0.6469
0.7222
0.7764
0.8164
0.8467
3
3.0
0.0054
0.2983
0.5323
0.6726
0.7597
0.8166
0.8557
0.8836
0.9041Table 3
OC values of sampling plans.
c
=
5
t
/
t
0.5
0
p
*
n
δ
0.01
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.75
137
0.1
0.2484
0.6314
0.8348
0.9234
0.9623
0.9802
0.9891
0.9936
0.9961
68
0.2
0.2493
0.6407
0.8427
0.9284
0.9652
0.9820
0.9901
0.9943
0.9966
34
0.4
0.2408
0.6506
0.8533
0.9354
0.9694
0.9845
0.9916
0.9952
0.9972
23
0.6
0.2228
0.6527
0.8596
0.9400
0.9723
0.9862
0.9926
0.9959
0.9976
17
0.8
0.2359
0.6888
0.8833
0.9527
0.9790
0.9898
0.9947
0.9971
0.9983
14
1.0
0.2120
0.6898
0.8883
0.9561
0.9809
0.9909
0.9953
0.9974
0.9985
10
1.5
0.1711
0.7193
0.9118
0.9685
0.9871
0.9942
0.9971
0.9985
0.9991
8
2.0
0.1556
0.7826
0.9428
0.9816
0.9930
0.9970
0.9986
0.9993
0.9996
7
2.5
0.1297
0.8341
0.9625
0.9889
0.9960
0.9983
0.9992
0.9996
0.9998
6
3.0
0.2323
0.9326
0.9880
0.9969
0.9989
0.9996
0.9998
0.9999
1.0000
0.90
171
0.1
0.0008
0.0143
0.0541
0.1154
0.1868
0.2595
0.3288
0.3925
0.4497
85
0.2
0.0007
0.0137
0.0536
0.1156
0.1879
0.2615
0.3314
0.3955
0.4531
42
0.4
0.0005
0.0129
0.0534
0.1173
0.1918
0.2673
0.3387
0.4038
0.4619
28
0.6
0.0004
0.0117
0.0519
0.1169
0.1930
0.2701
0.3428
0.4087
0.4675
21
0.8
0.0002
0.0108
0.0516
0.1185
0.1970
0.2760
0.3501
0.4170
0.4763
17
1.0
0.0001
0.0096
0.0501
0.1182
0.1983
0.2789
0.3542
0.4219
0.4818
11
1.5
0.0001
0.0120
0.0654
0.1499
0.2431
0.3321
0.4117
0.4810
0.5404
9
2.0
0.0000
0.0093
0.0619
0.1492
0.2459
0.3377
0.4193
0.4898
0.5499
8
2.5
0.0000
0.0078
0.0604
0.1503
0.2499
0.3439
0.4268
0.4980
0.5584
7
3.0
0.0000
0.0109
0.0779
0.1823
0.2907
0.3885
0.4722
0.5422
0.6006
0.95
193
0.1
0.0003
0.0071
0.0328
0.0787
0.1376
0.2019
0.2661
0.3273
0.3841
96
0.2
0.0002
0.0068
0.0324
0.0787
0.1382
0.2031
0.2680
0.3298
0.3869
47
0.4
0.0002
0.0067
0.0335
0.0824
0.1449
0.2124
0.2793
0.3425
0.4006
31
0.6
0.0001
0.0063
0.0339
0.0849
0.1499
0.2197
0.2884
0.3529
0.4117
23
0.8
0.0001
0.0062
0.0352
0.0892
0.1575
0.2301
0.3008
0.3666
0.4262
18
1.0
0.0001
0.0068
0.0395
0.0991
0.1728
0.2495
0.3228
0.3901
0.4503
12
1.5
0.0000
0.0070
0.0462
0.1166
0.2004
0.2845
0.3624
0.4320
0.4930
10
2.0
0.0000
0.0046
0.0399
0.1092
0.1938
0.2794
0.3590
0.4299
0.4920
8
2.5
0.0000
0.0078
0.0604
0.1503
0.2499
0.3439
0.4268
0.4980
0.5584
7
3.0
0.0000
0.0109
0.0779
0.1823
0.2907
0.3885
0.4722
0.5422
0.6006
0.99
240
0.1
0.0000
0.0015
0.0110
0.0339
0.0700
0.1153
0.1657
0.2177
0.2691
119
0.2
0.0000
0.0015
0.0109
0.0343
0.0710
0.1172
0.1683
0.2211
0.2731
58
0.4
0.0000
0.0015
0.0117
0.0369
0.0764
0.1254
0.1792
0.2341
0.2877
38
0.6
0.0000
0.0015
0.0122
0.0393
0.0813
0.1331
0.0189
0.0246
0.3012
28
0.8
0.0000
0.0015
0.0132
0.0428
0.0883
0.1435
0.2026
0.2617
0.3183
22
1.0
0.0000
0.0016
0.0149
0.0480
0.0978
0.1571
0.2195
0.2810
0.3392
15
1.5
0.0000
0.0014
0.0159
0.0535
0.1096
0.1750
0.2424
0.3075
0.3681
11
2.0
0.0000
0.0023
0.0256
0.0793
0.1518
0.2299
0.3057
0.3756
0.4383
9
2.5
0.0000
0.0034
0.0365
0.1055
0.1912
0.2781
0.3588
0.4307
0.4937
8
3.0
0.0000
0.0045
0.0457
0.1256
0.2197
0.3115
0.3945
0.4668
0.5289The producer’s risk is defined as the probability of rejecting the lot whentq>tq0. For a given value of the producer’s risk, say γ, we are interested in knowing the value of dq to ensure that the producer’s risk is less than or equal to γ if a sampling plan (n,c,t/tq) is developed at a specified confidence level p*. Thus, one needs to find the smallest value dq according to (10) as
(12)∑i=0c‍(ni)pi(1-p)n-i,
where
(13)p=F(ttq01dq).
To save space, based on the sampling plans (n,c,t/tq0) given in Table 1 the minimum ratios of d0.5 for the acceptability of a lot at the producer’s risk of γ=0.05 are presented in Table 4.Table 4
Minimum ratio for the acceptability of a lot with producer’s risk 0.05.
p
*
c
0.1
0.2
0.4
0.6
0.8
1.0
1.5
2.0
2.5
3.0
0.75
0
27.2995
27.3264
25.2820
25.3354
25.3887
31.7358
31.8683
42.4911
53.1138
32.2612
0.75
1
7.5211
7.4707
7.0638
6.9551
6.8415
7.0279
5.9340
7.9120
5.9356
7.1227
0.75
2
4.6982
4.6579
4.5740
4.4857
4.3925
4.1606
4.2240
4.2572
5.3215
4.2191
0.75
3
3.6690
3.6354
3.4860
3.5693
3.4898
3.5650
3.5335
3.0410
3.8013
3.1989
0.75
4
3.1398
3.1106
3.0490
2.9280
3.0224
2.9465
2.7276
3.0537
3.0610
2.6281
0.75
5
2.7959
2.7492
2.6938
2.6752
2.5684
2.5816
2.5929
2.5693
2.6240
2.3656
0.75
6
2.5649
2.5411
2.4575
2.4019
2.4074
2.3411
2.2305
2.2542
2.3369
2.8043
0.75
7
2.3985
2.3630
2.3429
2.2910
2.2887
2.1705
2.1968
2.3400
2.1320
2.5584
0.75
8
2.2725
2.2520
2.2076
2.1353
2.1038
2.1610
1.9808
2.1336
1.9782
2.3739
0.75
9
2.1635
2.1442
2.1022
2.0753
2.0432
2.0464
1.9775
1.9748
2.1740
2.2298
0.75
10
2.0847
2.0577
2.0176
1.9725
1.9931
1.9550
1.9712
1.8485
2.0464
2.1137
0.90
0
44.0826
44.1095
42.0653
44.2167
42.1723
42.2257
31.8683
42.4911
53.1138
63.7366
0.90
1
10.7007
10.4994
10.0945
9.6860
9.2735
10.0728
8.2474
7.9120
9.8900
7.1227
0.90
2
6.4092
6.2376
6.1553
6.0698
5.9809
5.4907
5.2368
5.6320
5.3215
6.3857
0.90
3
4.8111
4.7384
4.5906
4.5183
4.4422
4.3623
4.1426
3.8877
3.8013
4.5615
0.90
4
3.9865
3.9303
3.8153
3.7515
3.6839
3.5015
3.5805
3.6368
3.0610
3.6731
0.90
5
3.4961
3.4497
3.3544
3.2966
3.2348
3.2105
2.9167
3.0187
3.2117
3.1488
0.90
6
3.1546
3.0983
3.0488
2.9954
2.9380
2.8428
2.7494
2.6193
2.8177
2.8043
0.90
7
2.9122
2.8770
2.7767
2.6995
2.7270
2.5857
2.6252
2.6377
2.5401
2.5584
0.90
8
2.7309
2.6877
2.6215
2.5512
2.4762
2.5130
2.3481
2.3904
2.3338
2.3739
0.90
9
2.5801
2.5413
2.5003
2.4353
2.3657
2.3516
2.2950
2.2008
2.1740
2.2298
0.90
10
2.4598
2.4243
2.3678
2.3422
2.2768
2.2240
2.2503
2.2469
2.3107
2.1137
0.95
0
57.7180
56.6969
54.6527
50.5105
50.5640
52.7154
47.6037
42.4911
53.1138
63.7366
0.95
1
13.1233
12.9222
12.5182
12.4144
11.7015
11.5919
10.5419
10.9966
9.8900
11.8680
0.95
2
7.5278
7.4222
7.2090
6.8610
6.5094
6.8151
6.2408
5.6320
5.3215
6.3857
0.95
3
5.5593
5.4474
5.2215
5.2290
5.0756
4.7596
4.7466
4.7113
4.8596
4.5615
0.95
4
4.5600
4.4766
4.3623
4.2449
4.1238
4.0540
4.0012
3.6368
3.8171
3.6731
0.95
5
3.9491
3.9029
3.7669
3.6688
3.5670
3.4192
3.2373
3.4572
3.2117
3.1488
0.95
6
3.5477
3.4916
3.3770
3.2917
3.2025
3.1754
3.0050
2.9739
2.8177
2.8043
0.95
7
3.2501
3.2015
3.1018
3.0256
2.9455
2.8609
2.8366
2.6377
2.9250
3.0482
0.95
8
3.0289
2.9859
2.8972
2.8279
2.7544
2.7464
2.7084
2.6410
2.6670
2.8005
0.95
9
2.8479
2.7993
2.7389
2.6749
2.6067
2.5540
2.4517
2.4208
2.4685
2.6088
0.95
10
2.7041
2.6687
2.6127
2.5529
2.4889
2.4024
2.3882
2.2469
2.3107
2.4557
0.99
0
88.1383
86.0673
84.0232
81.9790
75.7389
73.6945
63.3385
63.4716
53.1138
63.7366
0.99
1
18.2712
17.7675
17.3645
16.9596
16.5526
16.1433
15.1091
14.0558
13.7457
11.8680
0.99
2
10.0941
9.9229
9.5790
9.2329
8.6206
8.7969
8.2360
8.3211
7.0400
8.4480
0.99
3
7.2133
7.1017
6.7979
6.6493
6.3403
6.3445
5.9462
5.5234
5.8892
5.8316
0.99
4
5.7617
5.6786
5.4559
5.2305
5.2214
4.8799
4.8365
4.7740
4.5460
4.5805
0.99
5
4.9170
4.8504
4.6741
4.5363
4.3954
4.2512
4.1877
3.8890
3.7734
3.8540
0.99
6
4.3502
4.2779
4.16472
3.9820
3.8621
3.8379
3.5116
3.6659
3.2741
3.3812
0.99
7
3.9395
3.8776
3.7517
3.6770
3.5993
3.4087
3.2557
3.2162
3.2971
3.0482
0.99
8
3.6363
3.5820
3.4483
3.3804
3.3092
3.2113
3.0646
2.8874
2.9880
3.2004
0.99
9
3.4034
3.3549
3.2557
3.1533
3.0872
3.0576
2.9161
2.8494
2.7510
2.9622
0.99
10
3.2100
3.1574
3.0673
2.9738
2.9118
2.8460
2.7972
2.6279
2.5624
2.7713Table 5
OC values of sampling plans forn=3, c=1, and δ0=2.0.
t
0.5
/
t
0.5
0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
OC
0.0832
0.4276
0.6317
0.7464
0.8157
0.8602
0.8905
0.9120
0.9272


## 4. Illustrative Examples

In this section, we consider two examples with real data sets to illustrate the proposed acceptance sampling plans.


### 4.1. Example 1

The first data refers to software reliability presented by Wood [20]. The data set was reported in hours as 519, 968, 1430, 1893, 2490, 3058, 3625, 4422, and 5218. The confidence level is assumed for this acceptance sampling plan only if the lifetimes are from half normal distribution with shape parameter σ. So in order to apply this example for our tables we have to confirm the goodness of fit of half normal distribution to the data in the example. This is done through the well-known QQ-plot method, and the value we get is R=0.9287. Hence we conclude that the half normal distribution provides a reasonable goodness of fit for the data.Suppose that the experimenter would like to establish the unknown 50th percentile life time for the software mentioned above to be at least 300 h and the life test would be ended at 600 h which should have led to the ratiot/t0.50=2.0. Thus, with c=1 and p*=0.75, the experimenter may take from Table 1 the sample size n which must be at least 3; thus the sampling plan to the data is given by
(14)(n,c,tt0.50)=(3,1,2.0).
Since there is one item with a failure time less than or equal to 600 h in the given sample of 9 observations, the lot is accepted as the result indicates that the 50th percentile life time t0.5 is at least 300 h with a confidence level of p*=0.75.The OC values for acceptance sampling plan in (14) and confidence p*=0.75, for a half normal distribution, may be taken from Table 2.The producer’s risk is almost equal to0.3683(=1-0.6317), when the true percentile is greater than or equal to 2.0 times the specified 50th percentile.From Table4, the experimenter could get the values of d0.5 for different choices of c and t/t0.50 in order to assert that the producer’s risk is less than 0.05. In this example, the value of d0.5 should be 7.9120 for c=1, t0.5/t0.50=2.0, and p*=0.75. This means that the product can have a life of 7.9120 times the required 50th percentile life time in order that under the above acceptance sampling plan the product is accepted with probability at least 0.75.


### 4.2. Example 2

The second data refers to the data obtained from Aarset [21]. It represents the lifetimes of 50 devices in hours. The data set was reported in hours as 0.1, 0.2, 1, 1, 1, 1, 1, 2, 3, 6, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85, 85, 86, and 86. On the similar lines of Example 1, for this data, the goodness of fit for 50 observations showed that R=0.9175.Suppose that we would like to establish the unknown 50th percentile life time for the devices to be at least 10 hours and the life test would be ended at 15 hrs which should have led to the ratio oft0.5/t0.50=1.5. Thus with c=5, p*=0.99, the experimenter may take, from Table 1, the sample size n which must be at least 15. Thus the sampling plan to the given data is given by (n,c,t/t0.50)=(15,5,1.5). Since there are 13 items with a failure time less than or equal to 15 hours in the given sample of 50 observations, the lot is accepted as the result indicates that the 50th percentile lifetime t0.5 is at least 10 hours with a confidence level of p*=0.99.


## 4.1. Example 1

The first data refers to software reliability presented by Wood [20]. The data set was reported in hours as 519, 968, 1430, 1893, 2490, 3058, 3625, 4422, and 5218. The confidence level is assumed for this acceptance sampling plan only if the lifetimes are from half normal distribution with shape parameter σ. So in order to apply this example for our tables we have to confirm the goodness of fit of half normal distribution to the data in the example. This is done through the well-known QQ-plot method, and the value we get is R=0.9287. Hence we conclude that the half normal distribution provides a reasonable goodness of fit for the data.Suppose that the experimenter would like to establish the unknown 50th percentile life time for the software mentioned above to be at least 300 h and the life test would be ended at 600 h which should have led to the ratiot/t0.50=2.0. Thus, with c=1 and p*=0.75, the experimenter may take from Table 1 the sample size n which must be at least 3; thus the sampling plan to the data is given by
(14)(n,c,tt0.50)=(3,1,2.0).
Since there is one item with a failure time less than or equal to 600 h in the given sample of 9 observations, the lot is accepted as the result indicates that the 50th percentile life time t0.5 is at least 300 h with a confidence level of p*=0.75.The OC values for acceptance sampling plan in (14) and confidence p*=0.75, for a half normal distribution, may be taken from Table 2.The producer’s risk is almost equal to0.3683(=1-0.6317), when the true percentile is greater than or equal to 2.0 times the specified 50th percentile.From Table4, the experimenter could get the values of d0.5 for different choices of c and t/t0.50 in order to assert that the producer’s risk is less than 0.05. In this example, the value of d0.5 should be 7.9120 for c=1, t0.5/t0.50=2.0, and p*=0.75. This means that the product can have a life of 7.9120 times the required 50th percentile life time in order that under the above acceptance sampling plan the product is accepted with probability at least 0.75.


## 4.2. Example 2

The second data refers to the data obtained from Aarset [21]. It represents the lifetimes of 50 devices in hours. The data set was reported in hours as 0.1, 0.2, 1, 1, 1, 1, 1, 2, 3, 6, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85, 85, 86, and 86. On the similar lines of Example 1, for this data, the goodness of fit for 50 observations showed that R=0.9175.Suppose that we would like to establish the unknown 50th percentile life time for the devices to be at least 10 hours and the life test would be ended at 15 hrs which should have led to the ratio oft0.5/t0.50=1.5. Thus with c=5, p*=0.99, the experimenter may take, from Table 1, the sample size n which must be at least 15. Thus the sampling plan to the given data is given by (n,c,t/t0.50)=(15,5,1.5). Since there are 13 items with a failure time less than or equal to 15 hours in the given sample of 50 observations, the lot is accepted as the result indicates that the 50th percentile lifetime t0.5 is at least 10 hours with a confidence level of p*=0.99.


## 5. Summary and Conclusions

This paper provides the minimum sample size required to decide upon accepting/rejecting a lot based on its specified 50th percentile (median) when the data follows half normal distribution. Assuming that the size of the given sample is minimum, we can use the tables in this paper to pick up acceptance numberc and the upper bound in order to accept/reject the lot for a given probability of acceptance, say p*. This paper also provides the sensitivity of the sampling plans in terms of the OC. We advise the industrial practitioner and the experimenter to adopt this plan in order to save the cost and time of the experiment. This plan can be further studied for many other distributions and various quality and reliability characteristics as future research.



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*Source: 302469-2013-12-04.xml*