68 Appendix D — Statistical Reference Tables

69 Appendix D — Statistical Reference Tables

This appendix provides essential statistical tables used throughout “AI-Powered Business Analytics” for hypothesis testing, confidence intervals, and critical value determination. All values are rounded to four decimal places unless otherwise noted.

69.1 1. Standard Normal Distribution Table

P(Z < z) — Cumulative Probability for the Standard Normal Distribution

This table shows the probability that a standard normal random variable is less than or equal to z.

Example: For z = 1.96, P(Z < 1.96) = 0.9750, meaning 97.50% of observations fall below 1.96 standard deviations above the mean.

69.1.1 Table: Standard Normal CDF

z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998

How to Use: - For P(Z < 1.96): Row 1.9, column .06 = 0.9750 - For P(Z > 1.96): 1 - 0.9750 = 0.0250 - For P(-1.96 < Z < 1.96): 0.9750 - 0.0250 = 0.9500 (95% confidence level)

69.2 2. t-Distribution Critical Values

t_{α/2,df} — Two-Tailed Critical Values for the t-Distribution

This table shows critical values for confidence intervals and hypothesis tests. For a two-tailed test at significance level α, use the column α. For a one-tailed test, use α/2.

df	α = 0.10	α = 0.05	α = 0.025	α = 0.01	α = 0.005
1	6.314	12.706	25.452	63.657	127.321
2	2.920	4.303	6.205	9.925	14.089
3	2.353	3.182	4.177	5.841	7.453
4	2.132	2.776	3.495	4.604	5.598
5	2.015	2.571	3.163	4.032	4.773
6	1.943	2.447	2.969	3.707	4.317
7	1.895	2.365	2.841	3.499	4.029
8	1.860	2.306	2.752	3.355	3.833
9	1.833	2.262	2.685	3.250	3.690
10	1.812	2.228	2.634	3.169	3.581
11	1.796	2.201	2.593	3.106	3.497
12	1.782	2.179	2.560	3.055	3.428
13	1.771	2.160	2.533	3.012	3.372
14	1.761	2.145	2.510	2.977	3.326
15	1.753	2.131	2.490	2.947	3.286
16	1.746	2.120	2.473	2.921	3.252
17	1.740	2.110	2.458	2.898	3.222
18	1.734	2.101	2.445	2.878	3.197
19	1.729	2.093	2.433	2.861	3.174
20	1.725	2.086	2.423	2.845	3.153
25	1.708	2.060	2.385	2.787	3.078
30	1.697	2.042	2.360	2.750	3.030
40	1.684	2.021	2.329	2.704	2.971
50	1.676	2.009	2.311	2.678	2.937
60	1.671	2.000	2.299	2.660	2.915
80	1.664	1.990	2.285	2.638	2.887
100	1.660	1.984	2.276	2.626	2.871
120	1.658	1.980	2.270	2.617	2.860
∞	1.645	1.960	1.960	2.576	2.807

How to Use: - 95% Confidence Interval for sample mean: Use df = n - 1, column α = 0.05. For df = 20, t = 2.086. - One-sample t-test at α = 0.05: Use df = n - 1, column α = 0.05.

69.3 3. Chi-Squared Distribution Critical Values

χ²_{α,df} — Right-Tail Critical Values for Chi-Squared Distribution

df	α = 0.995	α = 0.99	α = 0.975	α = 0.95	α = 0.90	α = 0.10	α = 0.05	α = 0.025	α = 0.01	α = 0.005
1	0.000	0.000	0.001	0.004	0.016	2.706	3.841	5.024	6.635	7.879
2	0.010	0.020	0.051	0.103	0.211	4.605	5.991	7.378	9.210	10.597
3	0.072	0.115	0.216	0.352	0.584	6.251	7.815	9.348	11.345	12.838
4	0.207	0.297	0.484	0.711	1.064	7.779	9.488	11.143	13.277	14.860
5	0.412	0.554	0.831	1.145	1.610	9.236	11.070	12.833	15.086	16.750
6	0.676	0.872	1.237	1.635	2.204	10.645	12.592	14.449	16.812	18.548
7	0.989	1.239	1.690	2.167	2.833	12.017	14.067	16.013	18.475	20.278
8	1.344	1.647	2.180	2.733	3.490	13.362	15.507	17.535	20.090	21.955
9	1.735	2.088	2.700	3.325	4.168	14.684	16.919	19.023	21.666	23.589
10	2.156	2.558	3.247	3.940	4.865	15.987	18.307	20.483	23.209	25.188
15	4.601	5.229	6.262	7.261	8.547	22.307	25.000	27.488	30.578	32.801
20	7.434	8.260	9.591	10.851	12.443	28.412	31.410	34.170	37.566	40.100
25	10.520	11.524	13.120	14.611	16.473	34.382	37.652	40.646	44.314	46.928
30	13.787	14.954	16.791	18.493	20.599	40.256	43.773	46.979	50.892	53.672

How to Use: - Goodness-of-fit test at α = 0.05: Find df = (number of categories - 1). Compare your test statistic to the critical value in the “α = 0.05” column. - For df = 5, α = 0.05: Critical value = 11.070. If your χ² statistic > 11.070, reject null hypothesis.

69.4 4. F-Distribution Critical Values (α = 0.05)

F_{0.05}(df1, df2) — Critical values for F-distribution at 5% significance level

df2 df1	1	2	3	4	5	6	7	8	9	10
1	161.4	199.5	215.7	224.6	230.2	234.0	236.8	238.9	240.5	241.9
2	18.51	19.00	19.16	19.25	19.30	19.33	19.35	19.37	19.38	19.40
5	6.61	5.79	5.41	5.19	5.05	4.95	4.88	4.82	4.77	4.74
10	4.96	4.10	3.71	3.48	3.33	3.22	3.14	3.07	3.02	2.98
20	4.35	3.49	3.10	2.87	2.71	2.60	2.51	2.45	2.39	2.35
30	4.17	3.32	2.92	2.69	2.53	2.42	2.33	2.27	2.21	2.16
60	4.00	3.15	2.76	2.53	2.37	2.25	2.17	2.10	2.04	1.99
120	3.92	3.07	2.68	2.45	2.29	2.18	2.09	2.02	1.96	1.91
∞	3.84	3.00	2.61	2.37	2.21	2.10	2.01	1.94	1.88	1.83

How to Use: - One-way ANOVA with 3 groups (df1 = 2) and 30 observations total (df2 = 27): - Interpolate: df2 = 27 is between 20 and 30. Use approximately 3.37. - If F > 3.37, reject null hypothesis of equal group means.

69.5 5. Durbin-Watson Test Critical Values

Durbin-Watson d Statistic — For Testing Autocorrelation in Regression Residuals

n	dL (α = 0.05)	dU (α = 0.05)	dL (α = 0.01)	dU (α = 0.01)
15	1.08	1.36	0.95	1.54
20	1.20	1.41	1.10	1.66
25	1.29	1.45	1.21	1.73
30	1.35	1.49	1.29	1.78
40	1.44	1.54	1.39	1.86
50	1.50	1.59	1.46	1.92
100	1.65	1.69	1.63	1.97

Interpretation: - d < dL: Evidence of positive autocorrelation - dL ≤ d ≤ dU: Inconclusive test - dU < d < 4 - dU: No autocorrelation - 4 - dU ≤ d ≤ 4 - dL: Inconclusive - d > 4 - dL: Evidence of negative autocorrelation

69.6 6. Wilcoxon Signed-Rank Test Critical Values (One-Tailed, α = 0.05)

Critical values for the Wilcoxon Signed-Rank test (T statistic)

n	One-Tail (α=0.05)	Two-Tail (α=0.05)
5	0	—
6	2	0
7	3	2
8	5	3
9	8	5
10	10	8
15	25	20
20	52	43
25	89	75

How to Use: - Rank the absolute differences; sum ranks for positive and negative differences separately. - Use the smaller sum as your T statistic. - If T ≤ critical value, reject null hypothesis of no difference.

69.7 7. Mann-Whitney U Test Critical Values (Two-Tailed, α = 0.05)

Critical values for Mann-Whitney U statistic

n1 n2	5	6	7	8	9	10
5	4	5	6	8	9	11
6	5	8	10	12	14	16
7	6	10	13	15	18	20
8	8	12	15	18	21	24
9	9	14	18	21	24	27
10	11	16	20	24	27	31

How to Use: - Combine both samples and rank all observations. - Calculate U for each group. - If U ≤ critical value from table, reject null hypothesis of equal distributions.

69.8 Worked Examples

69.8.1 Example 1: Confidence Interval for a Mean (t-Distribution)

Problem: You have a sample of 25 loan interest rates (in percentage) with mean = 11.5 and standard deviation = 2.1. Construct a 95% confidence interval.

Solution: 1. Find t-critical: df = 25 - 1 = 24; look up t_{0.05/2, 24}. From table, t = 2.064 2. Standard error: SE = 2.1 / √25 = 0.42 3. Margin of error: ME = 2.064 × 0.42 = 0.867 4. 95% CI: 11.5 ± 0.867 = [10.63, 12.37]

69.8.2 Example 2: Chi-Squared Test for Independence

Problem: Test whether customer satisfaction (satisfied vs. not satisfied) is independent of customer region (4 regions) at α = 0.05.

Solution: 1. Degrees of freedom: df = (2 - 1) × (4 - 1) = 3 2. Find critical value: From table, χ²_{0.05, 3} = 7.815 3. Calculate test statistic from observed and expected frequencies: χ² = 12.34 4. Decision: 12.34 > 7.815, so reject null hypothesis. Satisfaction and region are associated.

69.8.3 Example 3: ANOVA F-Test

Problem: Compare average sales across 4 sales teams with total sample size n = 40. Test at α = 0.05.

Solution: 1. df1 = 4 - 1 = 3 (numerator); df2 = 40 - 4 = 36 (denominator) 2. Look up F-critical: Interpolating between df2 = 30 and ∞, F_{0.05}(3, 36) ≈ 2.86 3. If your calculated F > 2.86, reject null hypothesis of equal team means.

69.9 References and Notes

All values are from standard statistical tables and match commonly used references.
For values not in these tables, use statistical software (R, Python, Excel) with functions like qnorm(), qt(), qchisq(), qf().
For very large sample sizes, the normal approximation (z-table) can replace t-distribution values.
One-tailed tests use half the two-tailed α value.

For more detailed tables and extended ranges, consult standard statistical references or use software like R or Python.

--- title: "Appendix D — Statistical Reference Tables" --- # Appendix D — Statistical Reference Tables This appendix provides essential statistical tables used throughout "AI-Powered Business Analytics" for hypothesis testing, confidence intervals, and critical value determination. All values are rounded to four decimal places unless otherwise noted. ## 1. Standard Normal Distribution Table **P(Z < z) — Cumulative Probability for the Standard Normal Distribution** This table shows the probability that a standard normal random variable is less than or equal to z. **Example**: For z = 1.96, P(Z < 1.96) = 0.9750, meaning 97.50% of observations fall below 1.96 standard deviations above the mean. ### Table: Standard Normal CDF | z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 | |---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----| | 0.0 | 0.5000 | 0.5040 | 0.5080 | 0.5120 | 0.5160 | 0.5199 | 0.5239 | 0.5279 | 0.5319 | 0.5359 | | 0.1 | 0.5398 | 0.5438 | 0.5478 | 0.5517 | 0.5557 | 0.5596 | 0.5636 | 0.5675 | 0.5714 | 0.5753 | | 0.2 | 0.5793 | 0.5832 | 0.5871 | 0.5910 | 0.5948 | 0.5987 | 0.6026 | 0.6064 | 0.6103 | 0.6141 | | 0.3 | 0.6179 | 0.6217 | 0.6255 | 0.6293 | 0.6331 | 0.6368 | 0.6406 | 0.6443 | 0.6480 | 0.6517 | | 0.4 | 0.6554 | 0.6591 | 0.6628 | 0.6664 | 0.6700 | 0.6736 | 0.6772 | 0.6808 | 0.6844 | 0.6879 | | 0.5 | 0.6915 | 0.6950 | 0.6985 | 0.7019 | 0.7054 | 0.7088 | 0.7123 | 0.7157 | 0.7190 | 0.7224 | | 0.6 | 0.7257 | 0.7291 | 0.7324 | 0.7357 | 0.7389 | 0.7422 | 0.7454 | 0.7486 | 0.7517 | 0.7549 | | 0.7 | 0.7580 | 0.7611 | 0.7642 | 0.7673 | 0.7704 | 0.7734 | 0.7764 | 0.7794 | 0.7823 | 0.7852 | | 0.8 | 0.7881 | 0.7910 | 0.7939 | 0.7967 | 0.7995 | 0.8023 | 0.8051 | 0.8078 | 0.8106 | 0.8133 | | 0.9 | 0.8159 | 0.8186 | 0.8212 | 0.8238 | 0.8264 | 0.8289 | 0.8315 | 0.8340 | 0.8365 | 0.8389 | | 1.0 | 0.8413 | 0.8438 | 0.8461 | 0.8485 | 0.8508 | 0.8531 | 0.8554 | 0.8577 | 0.8599 | 0.8621 | | 1.1 | 0.8643 | 0.8665 | 0.8686 | 0.8708 | 0.8729 | 0.8749 | 0.8770 | 0.8790 | 0.8810 | 0.8830 | | 1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 | 0.8944 | 0.8962 | 0.8980 | 0.8997 | 0.9015 | | 1.3 | 0.9032 | 0.9049 | 0.9066 | 0.9082 | 0.9099 | 0.9115 | 0.9131 | 0.9147 | 0.9162 | 0.9177 | | 1.4 | 0.9192 | 0.9207 | 0.9222 | 0.9236 | 0.9251 | 0.9265 | 0.9279 | 0.9292 | 0.9306 | 0.9319 | | 1.5 | 0.9332 | 0.9345 | 0.9357 | 0.9370 | 0.9382 | 0.9394 | 0.9406 | 0.9418 | 0.9429 | 0.9441 | | 1.6 | 0.9452 | 0.9463 | 0.9474 | 0.9484 | 0.9495 | 0.9505 | 0.9515 | 0.9525 | 0.9535 | 0.9545 | | 1.7 | 0.9554 | 0.9564 | 0.9573 | 0.9582 | 0.9591 | 0.9599 | 0.9608 | 0.9616 | 0.9625 | 0.9633 | | 1.8 | 0.9641 | 0.9649 | 0.9656 | 0.9664 | 0.9671 | 0.9678 | 0.9686 | 0.9693 | 0.9699 | 0.9706 | | 1.9 | 0.9713 | 0.9719 | 0.9726 | 0.9732 | 0.9738 | 0.9744 | 0.9750 | 0.9756 | 0.9761 | 0.9767 | | 2.0 | 0.9772 | 0.9778 | 0.9783 | 0.9788 | 0.9793 | 0.9798 | 0.9803 | 0.9808 | 0.9812 | 0.9817 | | 2.1 | 0.9821 | 0.9826 | 0.9830 | 0.9834 | 0.9838 | 0.9842 | 0.9846 | 0.9850 | 0.9854 | 0.9857 | | 2.2 | 0.9861 | 0.9864 | 0.9868 | 0.9871 | 0.9875 | 0.9878 | 0.9881 | 0.9884 | 0.9887 | 0.9890 | | 2.3 | 0.9893 | 0.9896 | 0.9898 | 0.9901 | 0.9904 | 0.9906 | 0.9909 | 0.9911 | 0.9913 | 0.9916 | | 2.4 | 0.9918 | 0.9920 | 0.9922 | 0.9925 | 0.9927 | 0.9929 | 0.9931 | 0.9932 | 0.9934 | 0.9936 | | 2.5 | 0.9938 | 0.9940 | 0.9941 | 0.9943 | 0.9945 | 0.9946 | 0.9948 | 0.9949 | 0.9951 | 0.9952 | | 2.6 | 0.9953 | 0.9955 | 0.9956 | 0.9957 | 0.9959 | 0.9960 | 0.9961 | 0.9962 | 0.9963 | 0.9964 | | 2.7 | 0.9965 | 0.9966 | 0.9967 | 0.9968 | 0.9969 | 0.9970 | 0.9971 | 0.9972 | 0.9973 | 0.9974 | | 2.8 | 0.9974 | 0.9975 | 0.9976 | 0.9977 | 0.9977 | 0.9978 | 0.9979 | 0.9979 | 0.9980 | 0.9981 | | 2.9 | 0.9981 | 0.9982 | 0.9982 | 0.9983 | 0.9984 | 0.9984 | 0.9985 | 0.9985 | 0.9986 | 0.9986 | | 3.0 | 0.9987 | 0.9987 | 0.9987 | 0.9988 | 0.9988 | 0.9989 | 0.9989 | 0.9989 | 0.9990 | 0.9990 | | 3.1 | 0.9990 | 0.9991 | 0.9991 | 0.9991 | 0.9992 | 0.9992 | 0.9992 | 0.9992 | 0.9993 | 0.9993 | | 3.2 | 0.9993 | 0.9993 | 0.9994 | 0.9994 | 0.9994 | 0.9994 | 0.9994 | 0.9995 | 0.9995 | 0.9995 | | 3.3 | 0.9995 | 0.9995 | 0.9995 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 | | 3.4 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9998 | **How to Use**: - For P(Z < 1.96): Row 1.9, column .06 = 0.9750 - For P(Z > 1.96): 1 - 0.9750 = 0.0250 - For P(-1.96 < Z < 1.96): 0.9750 - 0.0250 = 0.9500 (95% confidence level) --- ## 2. t-Distribution Critical Values **t_{α/2,df}** — Two-Tailed Critical Values for the t-Distribution This table shows critical values for confidence intervals and hypothesis tests. For a two-tailed test at significance level α, use the column α. For a one-tailed test, use α/2. | df | α = 0.10 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 | |----|----------|----------|-----------|----------|-----------| | 1 | 6.314 | 12.706 | 25.452 | 63.657 | 127.321 | | 2 | 2.920 | 4.303 | 6.205 | 9.925 | 14.089 | | 3 | 2.353 | 3.182 | 4.177 | 5.841 | 7.453 | | 4 | 2.132 | 2.776 | 3.495 | 4.604 | 5.598 | | 5 | 2.015 | 2.571 | 3.163 | 4.032 | 4.773 | | 6 | 1.943 | 2.447 | 2.969 | 3.707 | 4.317 | | 7 | 1.895 | 2.365 | 2.841 | 3.499 | 4.029 | | 8 | 1.860 | 2.306 | 2.752 | 3.355 | 3.833 | | 9 | 1.833 | 2.262 | 2.685 | 3.250 | 3.690 | | 10 | 1.812 | 2.228 | 2.634 | 3.169 | 3.581 | | 11 | 1.796 | 2.201 | 2.593 | 3.106 | 3.497 | | 12 | 1.782 | 2.179 | 2.560 | 3.055 | 3.428 | | 13 | 1.771 | 2.160 | 2.533 | 3.012 | 3.372 | | 14 | 1.761 | 2.145 | 2.510 | 2.977 | 3.326 | | 15 | 1.753 | 2.131 | 2.490 | 2.947 | 3.286 | | 16 | 1.746 | 2.120 | 2.473 | 2.921 | 3.252 | | 17 | 1.740 | 2.110 | 2.458 | 2.898 | 3.222 | | 18 | 1.734 | 2.101 | 2.445 | 2.878 | 3.197 | | 19 | 1.729 | 2.093 | 2.433 | 2.861 | 3.174 | | 20 | 1.725 | 2.086 | 2.423 | 2.845 | 3.153 | | 25 | 1.708 | 2.060 | 2.385 | 2.787 | 3.078 | | 30 | 1.697 | 2.042 | 2.360 | 2.750 | 3.030 | | 40 | 1.684 | 2.021 | 2.329 | 2.704 | 2.971 | | 50 | 1.676 | 2.009 | 2.311 | 2.678 | 2.937 | | 60 | 1.671 | 2.000 | 2.299 | 2.660 | 2.915 | | 80 | 1.664 | 1.990 | 2.285 | 2.638 | 2.887 | | 100 | 1.660 | 1.984 | 2.276 | 2.626 | 2.871 | | 120 | 1.658 | 1.980 | 2.270 | 2.617 | 2.860 | | ∞ | 1.645 | 1.960 | 1.960 | 2.576 | 2.807 | **How to Use**: - **95% Confidence Interval for sample mean**: Use df = n - 1, column α = 0.05. For df = 20, t = 2.086. - **One-sample t-test at α = 0.05**: Use df = n - 1, column α = 0.05. --- ## 3. Chi-Squared Distribution Critical Values **χ²_{α,df}** — Right-Tail Critical Values for Chi-Squared Distribution | df | α = 0.995 | α = 0.99 | α = 0.975 | α = 0.95 | α = 0.90 | α = 0.10 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 | |----|-----------|----------|-----------|----------|----------|----------|----------|-----------|----------|-----------| | 1 | 0.000 | 0.000 | 0.001 | 0.004 | 0.016 | 2.706 | 3.841 | 5.024 | 6.635 | 7.879 | | 2 | 0.010 | 0.020 | 0.051 | 0.103 | 0.211 | 4.605 | 5.991 | 7.378 | 9.210 | 10.597 | | 3 | 0.072 | 0.115 | 0.216 | 0.352 | 0.584 | 6.251 | 7.815 | 9.348 | 11.345 | 12.838 | | 4 | 0.207 | 0.297 | 0.484 | 0.711 | 1.064 | 7.779 | 9.488 | 11.143 | 13.277 | 14.860 | | 5 | 0.412 | 0.554 | 0.831 | 1.145 | 1.610 | 9.236 | 11.070 | 12.833 | 15.086 | 16.750 | | 6 | 0.676 | 0.872 | 1.237 | 1.635 | 2.204 | 10.645 | 12.592 | 14.449 | 16.812 | 18.548 | | 7 | 0.989 | 1.239 | 1.690 | 2.167 | 2.833 | 12.017 | 14.067 | 16.013 | 18.475 | 20.278 | | 8 | 1.344 | 1.647 | 2.180 | 2.733 | 3.490 | 13.362 | 15.507 | 17.535 | 20.090 | 21.955 | | 9 | 1.735 | 2.088 | 2.700 | 3.325 | 4.168 | 14.684 | 16.919 | 19.023 | 21.666 | 23.589 | | 10 | 2.156 | 2.558 | 3.247 | 3.940 | 4.865 | 15.987 | 18.307 | 20.483 | 23.209 | 25.188 | | 15 | 4.601 | 5.229 | 6.262 | 7.261 | 8.547 | 22.307 | 25.000 | 27.488 | 30.578 | 32.801 | | 20 | 7.434 | 8.260 | 9.591 | 10.851 | 12.443 | 28.412 | 31.410 | 34.170 | 37.566 | 40.100 | | 25 | 10.520 | 11.524 | 13.120 | 14.611 | 16.473 | 34.382 | 37.652 | 40.646 | 44.314 | 46.928 | | 30 | 13.787 | 14.954 | 16.791 | 18.493 | 20.599 | 40.256 | 43.773 | 46.979 | 50.892 | 53.672 | **How to Use**: - **Goodness-of-fit test at α = 0.05**: Find df = (number of categories - 1). Compare your test statistic to the critical value in the "α = 0.05" column. - **For df = 5, α = 0.05**: Critical value = 11.070. If your χ² statistic > 11.070, reject null hypothesis. --- ## 4. F-Distribution Critical Values (α = 0.05) **F_{0.05}(df1, df2)** — Critical values for F-distribution at 5% significance level | df2 \ df1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |-----------|---|---|---|---|---|---|---|---|---|----| | 1 | 161.4 | 199.5 | 215.7 | 224.6 | 230.2 | 234.0 | 236.8 | 238.9 | 240.5 | 241.9 | | 2 | 18.51 | 19.00 | 19.16 | 19.25 | 19.30 | 19.33 | 19.35 | 19.37 | 19.38 | 19.40 | | 5 | 6.61 | 5.79 | 5.41 | 5.19 | 5.05 | 4.95 | 4.88 | 4.82 | 4.77 | 4.74 | | 10 | 4.96 | 4.10 | 3.71 | 3.48 | 3.33 | 3.22 | 3.14 | 3.07 | 3.02 | 2.98 | | 20 | 4.35 | 3.49 | 3.10 | 2.87 | 2.71 | 2.60 | 2.51 | 2.45 | 2.39 | 2.35 | | 30 | 4.17 | 3.32 | 2.92 | 2.69 | 2.53 | 2.42 | 2.33 | 2.27 | 2.21 | 2.16 | | 60 | 4.00 | 3.15 | 2.76 | 2.53 | 2.37 | 2.25 | 2.17 | 2.10 | 2.04 | 1.99 | | 120 | 3.92 | 3.07 | 2.68 | 2.45 | 2.29 | 2.18 | 2.09 | 2.02 | 1.96 | 1.91 | | ∞ | 3.84 | 3.00 | 2.61 | 2.37 | 2.21 | 2.10 | 2.01 | 1.94 | 1.88 | 1.83 | **How to Use**: - **One-way ANOVA with 3 groups (df1 = 2) and 30 observations total (df2 = 27)**: - Interpolate: df2 = 27 is between 20 and 30. Use approximately 3.37. - If F > 3.37, reject null hypothesis of equal group means. --- ## 5. Durbin-Watson Test Critical Values **Durbin-Watson d Statistic** — For Testing Autocorrelation in Regression Residuals | n | dL (α = 0.05) | dU (α = 0.05) | dL (α = 0.01) | dU (α = 0.01) | |---|---------------|---------------|---------------|---------------| | 15 | 1.08 | 1.36 | 0.95 | 1.54 | | 20 | 1.20 | 1.41 | 1.10 | 1.66 | | 25 | 1.29 | 1.45 | 1.21 | 1.73 | | 30 | 1.35 | 1.49 | 1.29 | 1.78 | | 40 | 1.44 | 1.54 | 1.39 | 1.86 | | 50 | 1.50 | 1.59 | 1.46 | 1.92 | | 100 | 1.65 | 1.69 | 1.63 | 1.97 | **Interpretation**: - **d < dL**: Evidence of positive autocorrelation - **dL ≤ d ≤ dU**: Inconclusive test - **dU < d < 4 - dU**: No autocorrelation - **4 - dU ≤ d ≤ 4 - dL**: Inconclusive - **d > 4 - dL**: Evidence of negative autocorrelation --- ## 6. Wilcoxon Signed-Rank Test Critical Values (One-Tailed, α = 0.05) **Critical values for the Wilcoxon Signed-Rank test (T statistic)** | n | One-Tail (α=0.05) | Two-Tail (α=0.05) | |---|---|---| | 5 | 0 | — | | 6 | 2 | 0 | | 7 | 3 | 2 | | 8 | 5 | 3 | | 9 | 8 | 5 | | 10 | 10 | 8 | | 15 | 25 | 20 | | 20 | 52 | 43 | | 25 | 89 | 75 | **How to Use**: - Rank the absolute differences; sum ranks for positive and negative differences separately. - Use the smaller sum as your T statistic. - If T ≤ critical value, reject null hypothesis of no difference. --- ## 7. Mann-Whitney U Test Critical Values (Two-Tailed, α = 0.05) **Critical values for Mann-Whitney U statistic** | n1 \ n2 | 5 | 6 | 7 | 8 | 9 | 10 | |---------|---|---|---|---|---|----| | 5 | 4 | 5 | 6 | 8 | 9 | 11 | | 6 | 5 | 8 | 10 | 12 | 14 | 16 | | 7 | 6 | 10 | 13 | 15 | 18 | 20 | | 8 | 8 | 12 | 15 | 18 | 21 | 24 | | 9 | 9 | 14 | 18 | 21 | 24 | 27 | | 10 | 11 | 16 | 20 | 24 | 27 | 31 | **How to Use**: - Combine both samples and rank all observations. - Calculate U for each group. - If U ≤ critical value from table, reject null hypothesis of equal distributions. --- ## Worked Examples ### Example 1: Confidence Interval for a Mean (t-Distribution) **Problem**: You have a sample of 25 loan interest rates (in percentage) with mean = 11.5 and standard deviation = 2.1. Construct a 95% confidence interval. **Solution**: 1. Find t-critical: df = 25 - 1 = 24; look up t_{0.05/2, 24}. From table, t = 2.064 2. Standard error: SE = 2.1 / √25 = 0.42 3. Margin of error: ME = 2.064 × 0.42 = 0.867 4. **95% CI: 11.5 ± 0.867 = [10.63, 12.37]** ### Example 2: Chi-Squared Test for Independence **Problem**: Test whether customer satisfaction (satisfied vs. not satisfied) is independent of customer region (4 regions) at α = 0.05. **Solution**: 1. Degrees of freedom: df = (2 - 1) × (4 - 1) = 3 2. Find critical value: From table, χ²_{0.05, 3} = 7.815 3. Calculate test statistic from observed and expected frequencies: χ² = 12.34 4. **Decision**: 12.34 > 7.815, so reject null hypothesis. Satisfaction and region are associated. ### Example 3: ANOVA F-Test **Problem**: Compare average sales across 4 sales teams with total sample size n = 40. Test at α = 0.05. **Solution**: 1. df1 = 4 - 1 = 3 (numerator); df2 = 40 - 4 = 36 (denominator) 2. Look up F-critical: Interpolating between df2 = 30 and ∞, F_{0.05}(3, 36) ≈ 2.86 3. If your calculated F > 2.86, reject null hypothesis of equal team means. --- ## References and Notes - All values are from standard statistical tables and match commonly used references. - For values not in these tables, use statistical software (R, Python, Excel) with functions like `qnorm()`, `qt()`, `qchisq()`, `qf()`. - For very large sample sizes, the normal approximation (z-table) can replace t-distribution values. - One-tailed tests use half the two-tailed α value. --- *For more detailed tables and extended ranges, consult standard statistical references or use software like R or Python.*