Approximate Calculation Method for Noncentral t-Distribution Quantile

Table 5

Corollary results of coefficient k (C = 0.9).

	0.7
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	2.227727	1.95304	−12.33	3.927345	76.29	2.172989	−2.46
4	1.692664	1.592182	−5.94	1.818662	7.44	1.627362	−3.86
5	1.455446	1.409432	−3.16	1.468263	0.88	1.406016	−3.4
6	1.317233	1.294781	−1.7	1.30651	−0.81	1.279076	−2.9
7	1.225071	1.214403	−0.87	1.208123	−1.38	1.19427	−2.51
8	1.15805	1.154134	−0.34	1.139942	−1.56	1.132505	−2.21
9	1.106482	1.106785	0.03	1.088974	−1.58	1.084946	−1.95
10	1.065516	1.068362	0.27	1.048945	−1.56	1.046877	−1.75
20	0.873674	0.88102	0.84	0.86496	−1	0.866203	−0.86
30	0.800592	0.806951	0.79	0.794931	−0.71	0.79604	−0.57
40	0.759473	0.764901	0.71	0.755368	−0.54	0.756278	−0.42
Error range			−12.33∼0.84		−1.58∼76.29		−3.86∼−0.42
	0.8
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	3.039117	2.738367	−9.9	5.814814	91.33	3.03893	−0.01
4	2.294540	2.186456	−4.71	2.534914	10.48	2.236538	−2.53
5	1.975691	1.92394	−2.62	2.023345	2.41	1.927019	−2.46
6	1.794452	1.766061	−1.58	1.797266	0.16	1.755281	−2.18
7	1.675329	1.658734	−0.99	1.664035	−0.67	1.643227	−1.92
8	1.589881	1.580105	−0.61	1.573872	−1.01	1.563058	−1.69
9	1.525238	1.519469	−0.38	1.507702	−1.15	1.502188	−1.51
10	1.474096	1.470974	−0.21	1.456488	−1.19	1.454012	−1.36
20	1.240591	1.243243	0.21	1.229623	−0.88	1.232126	−0.68
30	1.154152	1.15689	0.24	1.146775	−0.64	1.148946	−0.45
40	1.106253	1.108681	0.22	1.100748	−0.5	1.102528	−0.34
Error range			−9.9∼0.24		−1.19∼91.33		−2.53∼−0.01
	0.9
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	4.258037	3.827763	−10.11	8.552616	101.86	4.340567	1.94
4	3.187642	3.026395	−5.06	3.594739	12.77	3.144705	−1.35
5	2.742235	2.658306	−3.06	2.842316	3.65	2.698114	−1.61
6	2.493491	2.442339	−2.05	2.518136	0.99	2.456109	−1.5
7	2.332315	2.298158	−1.46	2.331059	−0.05	2.301023	−1.34
8	2.218358	2.193841	−1.11	2.206595	−0.53	2.191628	−1.2
9	2.132705	2.114345	−0.86	2.116515	−0.76	2.109516	−1.09
10	2.065574	2.051333	−0.69	2.047597	−0.87	2.045142	−0.99
20	1.765161	1.762608	−0.14	1.751624	−0.77	1.756306	−0.5
30	1.656991	1.656266	−0.04	1.647536	−0.57	1.651498	−0.33
40	1.597842	1.597666	−0.01	1.590631	−0.45	1.593871	−0.25
Error range			−10.11∼−0.01		−0.87∼101.86		−1.61∼1.94
	0.95
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	5.311158	4.75345	−10.5	10.85606	104.4	5.457418	2.75
4	3.955887	3.743654	−5.36	4.5002	13.76	3.924381	−0.8
5	3.399631	3.286259	−3.33	3.543087	4.22	3.35901	−1.19
6	3.0916004	3.020273	−2.31	3.134426	1.39	3.055638	−1.16
7	2.8934486	2.844041	−1.71	2.90053	0.24	2.862743	−1.06
8	2.7540745	2.717418	−1.33	2.746018	−0.29	2.727541	−0.96
9	2.6498059	2.62142	−1.07	2.634866	−0.56	2.626593	−0.88
10	2.568089	2.545597	−0.88	2.550266	−0.69	2.547805	−0.79
20	2.207616	2.201785	−0.26	2.19221	−0.7	2.19873	−0.4
30	2.079723	2.077018	−0.13	2.06865	−0.53	2.074132	−0.27
40	2.010191	2.008534	−0.08	2.00167	−0.42	2.006152	−0.2
Error range			−10.5∼−0.08		−0.7∼104.4		−1.19∼2.75