Approximate Calculation Method for Noncentral t-Distribution Quantile

Table 4

Corollary results of coefficient k (C = 0.8).

	0.7
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	1.441459	1.198475	−16.86	2.240555	55.44	1.392952	−3.37
4	1.199003	1.056408	−11.89	1.233037	2.84	1.171917	−2.26
5	1.079699	0.980532	−9.18	1.065791	−1.29	1.06182	−1.66
6	1.006262	0.931047	−7.47	0.984826	−2.13	0.993137	−1.3
7	0.955358	0.895292	−6.29	0.933649	−2.27	0.945115	−1.07
8	0.917479	0.867706	−5.42	0.897132	−2.22	0.909135	−0.91
9	0.887813	0.845589	−4.76	0.869215	−2.09	0.880892	−0.78
10	0.863964	0.827366	−4.24	0.846899	−1.98	0.857968	−0.69
20	0.748168	0.733359	−1.98	0.739788	−1.12	0.745878	−0.31
30	0.70254	0.693546	−1.28	0.697122	−0.77	0.70116	−0.2
40	0.676569	0.670218	−0.94	0.672596	−0.59	0.675594	−0.14
Error range			−16.86∼−0.94		−2.27∼55.44		−3.37∼−0.14
	0.8
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	2.015963	1.864198	−7.53	3.350253	66.19	1.959245	−2.81
4	1.67476	1.595514	−4.73	1.756294	4.87	1.644344	−1.82
5	1.513834	1.461578	−3.45	1.508846	−0.33	1.494208	−1.3
6	1.417233	1.378831	−2.71	1.395374	−1.54	1.403193	−0.99
7	1.35164	1.321513	−2.23	1.326381	−1.87	1.340847	−0.8
8	1.303523	1.278875	−1.89	1.278522	−1.92	1.294852	−0.67
9	1.266389	1.245617	−1.64	1.242699	−1.87	1.259186	−0.57
10	1.236572	1.218831	−1.43	1.214529	−1.78	1.230522	−0.49
20	1.096332	1.089234	−0.65	1.084534	−1.08	1.094046	−0.21
30	1.042614	1.038382	−0.41	1.03481	−0.75	1.041283	−0.13
40	1.012475	1.009494	−0.29	1.00666	−0.57	1.011538	−0.09
Error range			−7.53∼−0.29		−1.92∼66.19		−2.81∼−0.09
	0.9
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	2.870411	2.734096	−4.75	4.95068	72.47	2.800163	−2.45
4	2.372481	2.313091	−2.5	2.520845	6.25	2.337578	−1.47
5	2.145211	2.11023	−1.63	2.153216	0.37	2.124036	−0.99
6	2.011888	1.988188	−1.18	1.989734	−1.1	1.997447	−0.72
7	1.922926	1.905291	−0.92	1.892856	−1.56	1.912164	−0.56
8	1.858496	1.844579	−0.75	1.827017	−1.69	1.850058	−0.45
9	1.809182	1.797828	−0.63	1.778539	−1.69	1.802401	−0.37
10	1.770182	1.760544	−0.54	1.740922	−1.65	1.764428	−0.33
20	1.58978	1.586399	−0.21	1.573196	−1.04	1.587908	−0.12
30	1.522643	1.520797	−0.12	1.511473	−0.73	1.521655	−0.06
40	1.485504	1.484164	−0.09	1.477065	−0.57	1.484813	−0.05
Error range			−4.75∼−0.09		−1.69∼72.47		−2.45∼−0.05
	0.95
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	3.603836	3.451824	−4.22	6.293786	74.64	3.519022	−2.35
4	2.968097	2.908834	−2	3.169819	6.8	2.928243	−1.34
5	2.682675	2.65053	−1.2	2.700191	0.65	2.659239	−0.87
6	2.5169789	2.496323	−0.82	2.493566	−0.93	2.501342	−0.62
7	2.4069464	2.392608	−0.6	2.372342	−1.44	2.395769	−0.46
8	2.3278784	2.31704	−0.47	2.29066	−1.6	2.319351	−0.37
9	2.2677779	2.259295	−0.37	2.230948	−1.62	2.261001	−0.3
10	2.220237	2.213298	−0.31	2.184893	−1.59	2.214701	−0.25
20	2.003516	2.001622	−0.09	1.982907	−1.03	2.001987	−0.08
30	1.924118	1.92322	−0.05	1.910046	−0.73	1.923348	−0.04
40	1.880421	1.879741	−0.04	1.869768	−0.57	1.879932	−0.03
Error range			−4.22∼−0.04		−1.62∼74.64		−2.35∼−0.03