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Question 4.DS.16: Two different fastener tool manufacturers have submitted the......

Two different fastener tool manufacturers have submitted their tools to the Dorf fastening tool lab for testing. The tool engineers are interested in evaluating the time required to tighten the bolt on a given test bracket. Data is collected on run-down times (from snug to fully tightened) given in milliseconds for the two different manufacturers. The Dorf engineers are interested in reviewing the analysis of both the mean and variance of the run-down times. The Dorf engineers perform 16 run-down tests on each of two representative tools and find the following for run-down times:

Tool 1: n_{1} = 16; \overline{x}_{1} = 67.7; S_{1} = 8.46
Tool 2: n_{2} = 16; \overline{x}_{2} = 67.4; S_{2} = 8.73

The Dorf engineers decide to:

A. Test the hypothesis that the variances are equal. Using α = 0.02.
B. Using the results of (A) above, they will test the hypothesis that the mean run-down times are equal. Using α = 0.02.

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Approach:

1.H_{0} : \sigma ^{2}_{1}=\sigma ^{2}_{2}  H_{A} : \sigma ^{2}_{1}≠\sigma ^{2}_{2} (Note: This results in a two-tailed distribution)

2. a =0.02; \frac{\alpha}{2}=0.01  \sigma Unknown—Use t test
3.From the data: S_{1} = 8.46     S_{2} = 8.73   \overline{x}_{1} = 67.7   \overline{x}_{2} = 67.4

4. F_{0}= S^{2}_{1}/S^{2}_{2}=(8.46)^{2}/(8.73)^{2}=0.939

5. n_{1}=16      n_{2}=16

6. ∂_{1}= ∂_{2}=16-1=15

From Table 4.7 the Dorf engineers find  F_{0.01,15,15}=3.52;   F_{0.99,15,15}=\frac{1}{F_{0.01,15,15}} =\frac{1}{3.52}=0.284

Displaying this data on a sketch reveals the following:

The data shown indicates that we do not reject H_{0}, and the test indicates that the variances of the output of the two tools are essentially equal. Now to determine the mean “rundown” time we must use the t statistic here since σ is unknown and n_{1} and n_{2} are small. If the mean run-down times are equal, we have for the t statistic:

7.  t_0=\frac{\bar{x}_1-\bar{x}_2}{s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} ; \partial=n_1+n_2-1 \text{ where } S_p^2=\frac{\left(n_1-1\right) S_1^2+\left(n_2-1\right) S_2^2}{n_1+n_2-2}\\ 8.  S_p^2=\frac{\left(n_1-1\right) S_1^2+\left(n_2-1\right) S_2^2}{n_1+n_2-2}=\frac{(16-1)(8.46)^2+(16-1)(8.73)^2}{16+16-2}=73.892 ; S_p=8.596\\ 9.  t_0=\frac{\bar{x}_1-\bar{x}_2}{S_P \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{67.7-67.4}{8.596\left[\frac{1}{16}+\frac{1}{16}\right]^{1 / 2}}=0.0987

From Table 4.3 we find t_{\alpha/2,(n_{1}+n_{2}-2}) =t_{0.01,30}=2.457

Displaying this data on a sketch reveals the following:

Note that the t test requires normality. However, small departures from normality have little effect on the t test.
Given the data, the Dorf engineers do not reject H_{0}, and therefore conclude that the run-down times of the tools from the two manufacturers are essentially equal.

TABLE 4.7
Percentiles of the F-Distribution F.95 (∂1 , ∂2 )
∂1 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞
∂2
1 39.86 49.5 58.59 56.88 57.24 58.2 58.91 59.44 59.86 60.19 60.71 61.22 61.74 62 62.26 62.58 62.79 63.06 68.33
2 8.53 9 9.16 9.24 9.29 9.33 9.35 9.37 9.38 9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.48 9.49
3 5.54 6.46 6.39 5.34 5.31 5.28 5.27 5.25 5.24 5.23 5.22 5.2 5.18 5.18 5.17 5.16 5.15 5.14 5.13
4 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94 3.92 3.9 3.87 3.84 3.88 3.82 3.8 3.79 3.78 3.76
5 4.06 3.78 3.62 3.52 3.45 3.4 3.37 3.34 3.32 3.3 3.27 3.24 3.21 3.19 3.17 3.15 3.14 3.12 3.1
6 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96 2.94 2.9 2.87 2.84 2.82 2.8 2.78 2.76 2.74 2.72
7 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72 2.7 2.67 2.68 2.59 2.58 2.56 2.54 2.51 2.49 2.47
8 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56 2.6 2.5 2.46 2.42 2.4 2.38 2.36 2.34 2.32 2.29
9 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44 2.42 2.38 2.34 2.3 2.28 2.25 2.23 2.21 2.18 2.16
10 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.88 2.35 2.32 2.28 2.24 2.2 2.18 2.16 2.13 2.11 2.08 2.06
11 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.3 2.27 2.25 2.21 2.17 2.12 2.1 2.08 2.05 2.03 2 1.97
12 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21 2.19 2.15 2.1 2.06 2.04 2.01 1.99 1.96 1.98 1.9
13 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.2 2.16 2.14 2.1 2.05 2.01 1.98 1.96 1.93 1.9 1.88 1.85
14 3.1 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12 2.1 2.05 2.01 1.96 1.94 1.91 1.89 1.86 1.83 1.8
15 3.07 2.7 2.49 2.36 2.27 2.21 2.16 2.12 2.09 2.06 2.02 1.97 1.92 1.9 1.87 1.85 1.82 1.79 1.76
16 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06 2.03 1.99 1.94 1.89 1.87 1.84 1.81 1.78 1.76 1.72
17 3.03 2.64 2.44 2.31 2.22 2.15 2.1 2.06 2.03 2 1.96 1.91 1.86 1.84 1.81 1.78 1.75 1.72 1.69
18 3.01 2.62 2.42 2.29 2.2 2.13 2.08 2.04 2 1.98 1.93 1.89 1.84 1.81 1.78 1.75 1.72 1.69 1.68
19 2.99 2.61 2.4 2.27 2.18 2.11 2.06 2.02 1.98 1.96 1.91 1.86 1.81 1.79 1.76 1.73 1.7 1.67 1.63
20 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2 1.96 1.94 1.89 1.84 1.79 1.77 1.74 1.71 1.68 1.64 1.61
21 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95 1.92 1.87 1.83 1.78 1.75 1.72 1.69 1.66 1.62 1.59
22 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93 1.9 1.86 1.81 1.76 1.73 1.7 1.67 1.64 1.6 1.57
23 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92 1.89 1.84 1.8 1.74 1.72 1.69 1.66 1.62 1.58 1.55
24 2.93 2.54 2.33 2.19 2.1 2.04 1.98 1.94 1.91 1.88 1.83 1.78 1.73 1.7 1.67 1.64 1.61 1.57 1.53
25 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89 1.87 1.82 1.77 1.72 1.69 1.66 1.63 1.59 1.56 1.52
26 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88 1.86 1.81 1.76 1.71 1.68 1.65 1.61 1.58 1.54 1.5
27 2.9 2.51 2.3 2.17 2.07 2 1.95 1.91 1.87 1.85 1.8 1.75 1.7 1.67 1.64 1.6 1.57 1.53 1.49
28 2.89 2.5 2.29 2.16 2.06 2 1.94 1.9 1.87 1.84 1.79 1.74 1.69 1.66 1.63 1.59 1.56 1.52 1.48
29 2.89 2.5 2.28 2.15 2.06 1.99 1.93 1.89 1.86 1.83 1.78 1.73 1.68 1.65 1.62 1.58 1.55 1.51 1.47
30 2.88 2.49 2.28 2.14 2.06 1.98 1.93 1.88 1.85 1.82 1.77 1.72 1.67 1.64 1.61 1.57 1.54 1.5 1.46
40 2.84 2.44 2.23 2.09 2 1.93 1.87 1.83 1.79 1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.47 1.42 1.38
60 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74 1.71 1.66 1.6 1.54 1.51 1.48 1.44 1.4 1.35 1.29
120 2.75 2.35 2.13 1.99 1.9 1.82 1.77 1.72 1.68 1.65 1.6 1.55 1.48 1.45 1.41 1.37 1.32 1.26 1.19
∞ 2.71 2.3 2.08 1.94 1.85 1.77 1.72 1.67 1.63 1.6 1.55 1.49 1.42 1.38 1.34 1.3 1.24 1.17 1
1 161.4 199.5 215.7 224.6 280.2 234 236.8 288.9 240.5 241.9 243.9 245.9 248 249.1 250.1 251.1 252.2 253.3 254.3
2 18.51 19 19.16 19.25 19.3 19.33 19.35 19.37 19.38 19.4 19.41 19.43 19.45 19.45 19.46 1..47 19.48 19.49 19.5
3 10.18 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.7 8.66 8.64 8.62 8.59 8.57 8.55 8.53
4 7.71 6.94 6.59 6.89 6.28 6.16 6.09 6.04 6 5.96 5.91 5.86 5.8 5.77 5.75 5.72 5.89 5.66 5.63
5 6.61 5.79 5.41 6.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.58 4.53 4.5 4.46 4.43 4.4 4.36
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.1 4.06 4 3.94 3.87 3.84 3.81 3.77 3.74 3.7 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.67 3.51 3.44 3.41 3.38 3.34 3.3 3.27 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.5 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93
9 5.12 4.28 3.86 3.68 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.9 2.86 2.83 2.79 2.75 2.71
10 4.96 4.1 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.7 2.66 2.62 2.58 2.54
11 4.84 3.98 3.59 3.36 3.2 3.09 3.01 2.95 2.9 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.4
12 4.75 3.89 3.49 3.26 3.11 3 2.91 2.85 2.8 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.3
13 4.67 3.81 3.41 3.18 3.08 2.92 2.88 2.77 2.71 2.67 2.6 2.53 2.46 2.42 2.38 2.34 2.3 2.25 2.21
14 4.6 3.74 3.34 3.11 2.96 2.85 2.76 2.7 2.65 2.6 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13
15 4.54 3.68 3.29 3.06 2.9 2.79 2.71 2.64 2.59 2.54 2.48 2.4 2.33 2.29 2.25 2.2 2.16 2.11 2.07
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01
17 4.45 3.59 3.2 2.96 2.81 2.7 2.61 2.55 2.49 2.46 2.38 2.31 2.23 2.19 2.16 2.1 2.06 2.01 1.96
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92
19 4.38 3.52 3.13 2.9 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88
20 4.35 3.49 3.1 2.87 2.71 2.6 2.51 2.45 2.39 2.35 2.28 2.2 2.12 2.08 2.04 1.99 1.95 1.9 1.84
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.1 2.05 2.01 1.96 1.92 1.87 1.81
22 4.3 3.44 3.05 2.82 2.66 2.55 2.46 2.4 2.34 2.3 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78
23 4.28 3.42 3.03 2.8 2.64 2.53 2.44 2.37 2.32 2.27 2.2 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76
24 4.26 3.4 3.01 2.78 2.62 2.51 2.42 2.36 2.3 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73
25 4.24 3.39 2.99 2.76 2.6 2.49 2.4 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71
26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.9 1.85 1.8 1.75 1.69
27 4.21 3.35 2.98 2.73 2.57 2.46 2.37 2.31 2.25 2.2 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67
28 4.2 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65
29 4.18 3.33 2.93 2.7 2.55 2.43 2.35 2.28 2.22 2.18 2.1 2.03 1.94 1.9 1.85 1.81 1.75 1.7 1.64
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51
60 4 3.15 2.76 2.53 2.37 2.26 2.17 2.1 2.04 1.99 1.92 1.84 1.75 1.7 1.65 1.59 1.53 1.47 1.39
120 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.5 1.43 1.35 1.25
∞ 3.34 3 2.6 2.37 2.21 2.1 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1
1 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 976.7 984.9 993.1 997.2 1001 1006 1010 1014 1018
2 38.51 39 39.17 39.25 89.3 39.33 39.36 39.37 39.39 39.4 39.41 39.43 89.45 39.46 39.46 39.47 39.48 39.49 39.5
3 17.44 16.04 15.44 15.1 14.88 14.73 14.62 14.54 14.47 14.42 14.34 14.25 14.17 14.12 14.08 14.04 13.99 13.95 13.9
4 12.22 10.65 9.98 9.6 9.36 9.2 9.07 8.98 8.9 8.84 8.75 8.66 8.56 8.51 8.46 8.41 8.36 8.31 8.26
5 10.01 8.43 7,76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.52 6.43 6.38 6.28 6.23 6.18 6.12 6.07 6.02
6 8.81 7.26 6.6 6.23 5.99 5.82 5.7 5.6 5.52 5.46 5.37 5.27 5.17 5.12 5.07 5.01 4.96 4.9 4.85
7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.9 4.82 4.76 4.67 4.57 4.47 4.42 4.36 4.31 4.25 4.2 4.14
8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.3 4.2 4.1 4 3.95 3.89 3.84 3.73 3.73 3.67
9 7.21 5.71 5.08 4.72 4.48 4.32 4.2 4.1 4.08 3.96 3.87 3.77 3.67 3.61 3.56 3.51 3.45 3.39 3.33
10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.62 3.52 3.42 3.37 3.31 3.26 3.29 3.14 3.08
11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.69 3.58 3.48 3.33 3.23 3.17 3.12 3.06 3 2.94 2.88
12 6.55 5.1 4.47 4.12 3.89 3.78 3.61 3.61 3.44 3.37 3.28 3.18 3.07 3.02 2.96 2.91 2.85 2.79 2.72
13 6.41 4.97 4.35 4 3.77 3.6 3.48 3.39 3.31 3.25 3.15 3.05 2.95 2.89 2.84 2.78 2.72 2.66 2.6
14 6.3 4.86 4.24 3.89 3.66 3.5 3.38 3.29 3.21 3.15 3.05 2.95 2.84 2.79 2.73 2.67 2.61 2.55 2.49
15 6.2 4.77 4.15 3.8 3.58 3.41 3.29 3.2 3.12 3.06 2.98 2.86 2.76 2.7 2.64 2.59 2.52 2.46 2.4
16 6.12 4.69 4.08 3.73 3.5 3.34 3.22 3.12 3.05 2.99 2.89 2.79 2.68 2.63 2.57 2.51 2.45 2.38 2.32
17 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98 2.92 2.82 2.72 2.62 2.56 2.5 2.44 2.38 2.32 2.25
18 5.98 4.56 3.95 3.61 3.38 3.22 3.1 3.01 2.93 2.87 2.77 2.67 2.56 2.5 2.44 2.38 2.32 2.26 2.19
19 5.92 4.51 3.9 3.56 3.33 3.17 3.05 2.96 2.88 2.82 2.72 2.62 2.51 2.45 2.39 2.33 2.27 2.2 2.13
20 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.68 2.57 2.46 2.41 2.35 2.29 2.22 2.16 2.09
21 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.8 2.73 2.64 2.53 2.42 2.37 2.31 2.25 2.18 2.11 2.04
22 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76 2.7 2.6 2.5 2.39 2.33 2.27 2.21 2.14 2.08 2
23 5.75 4.35 3.75 3.41 3.18 3.02 2.9 2.81 2.73 2.67 2.57 2.47 2.36 2.3 2.24 2.18 2.11 2.04 1.97
24 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.7 2.64 2.54 2.44 2.33 2.27 2.21 2.15 2.08 2.01 1.94
25 5.69 4.29 3.59 3.35 3.13 2.97 2.85 2.75 2.68 2.61 2.51 2.41 2.3 2.24 2.18 2.12 2.05 1.98 1.91
26 5.66 4.27 3.67 3.33 3.1 2.94 2.82 2.73 2.65 2.59 2.49 2.39 2.28 2.22 2.16 2.09 2.03 1.95 1.88
27 5.63 4.24 3.65 3.31 3.08 2.92 2.8 2.71 2.63 2.57 2.47 2.36 2.25 2.19 2.13 2.07 2 1.93 1.85
28 5.61 4.22 3.63 3.29 3.06 2.9 2.78 2.68 2.61 2.55 2.45 2.34 2.23 2.17 2.11 2.05 1.98 1.91 1.83
29 5.59 4.2 3.61 3.27 3.04 2.88 2.76 2.67 2.59 2.53 2.43 2.32 2.21 2.15 2.09 2.03 1.96 1.89 1.81
30 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.51 2.41 2.31 2.2 2.14 2.07 2.01 1.94 1.87 1.79
40 5.42 4.05 3.46 3.13 2.9 2.74 2.62 2.53 2.45 2.39 2.29 2.18 2.07 2.01 1.94 1.88 1.8 1.72 1.64
60 5.29 3.93 3.34 3.01 2.79 2.68 2.51 2.41 2.33 2.27 2.17 2.06 1.94 1.88 1.82 1.74 1.67 1.58 1.48
120 5.15 3.8 3.23 2.89 2.67 2.52 2.39 2.3 2.22 2.16 2.05 1.94 1.82 1.76 1.69 1.61 1.58 1.43 1.31
∞ 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.94 1.88 1.71 1.64 1.57 1.48 1.39 1.27 1
1 4052 4999.6 5403 5625 5764 5859 5928 5982 6022 6056 6106 6157 6209 6235 6261 6287 6313 6339 6366
2 98.5 99 99.17 99.25 99.3 99.33 99.36 99.37 99.39 99.4 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.5
3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.87 26.69 26.6 26.59 26.41 26.32 26.22 26.13
4 21.2 18 16.69 15.98 15.52 15.21 14.98 14.8 14.66 14.55 14.37 14.2 14.02 13.93 13.84 13.75 13.65 13.56 13.46
5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.2 9.11 8.02
6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.1 7.98 7.87 7.72 7.56 7.4 7.31 7.23 7.14 7.06 6.97 6.88
7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65
8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.28 5.2 5.12 5.03 4.95 4.86
9 10.56 8.02 6.99 6.42 6.06 5.89 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.4 4.31
10 10.04 7.56 6.55 5.99 5.64 5.39 5.2 5.06 4.94 4.85 4.71 4.56 4.41 4.38 4.25 4.17 4.08 4 3.91
11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.4 4.25 4.1 4.02 3.94 3.86 3.78 3.69 3.6
12 9.33 6.98 5.95 5.41 5.06 4.82 4.64 4.5 4.39 4.3 4.16 4.01 3.86 3.78 3.7 3.62 3.54 3.45 3.36
13 9.07 6.7 5.74 5.21 4.86 4.62 4.44 4.3 4.19 4.1 3.98 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17
14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.08 3.94 3.8 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3
15 8.68 6.86 5.42 4.89 4.56 4.32 4.14 4 3.89 3.8 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87
16 8.53 6.28 5.29 4.77 4.44 4.2 4.03 3.89 3.78 3.69 3.55 3.41 3.26 3.18 3.1 3.02 2.93 2.84 2.75
17 8.4 6.11 5.18 4.67 4.34 4.1 3.93 3.79 3.68 3.59 3.46 3.31 3.16 3.08 3 2.92 2.83 2.75 2.65
18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.6 3.51 3.37 3.23 3.08 3 2.92 2.84 2.75 2.66 2.57
19 8.18 5.98 5.01 4.5 4.17 3.94 3.77 3.63 3.52 3.48 3.3 3.15 3 2.92 2.84 2.76 2.67 2.58 2.49
20 8.1 5.85 4.94 4.43 4.1 3.87 3.7 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42
21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.4 3.31 3.17 3.03 2.88 2.8 2.72 2.64 2.55 2.46 2.36
22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.5 2.4 2.31
23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.3 3.21 3.07 2.93 2.78 2.7 2.62 2.54 2.45 2.35 2.26
24 7.82 5.61 4.72 4.22 3.9 3.67 3.5 3.36 3.26 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.4 2.31 2.21
25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13 2.99 2.85 2.7 2.62 2.54 2.45 2.38 2.27 2.17
26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 3.09 2.96 2.81 2.66 2.58 2.5 2.42 2.33 2.23 2.13
27 7.68 5.49 4.6 4.11 3.78 3.56 3.39 3.26 3.15 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.2 2.1
28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 3.03 2.9 2.75 2.6 2.52 2.44 2.35 2.26 2.17 2.06
29 7.6 5.42 4.54 4.04 3.73 3.5 3.33 3.2 3.09 3 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03
30 7.56 5.39 4.51 4.02 3.7 3.47 3.3 3.17 3.07 2.98 2.84 2.7 2.55 2.47 2.39 2.3 2.21 2.11 2.01
40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.8 2.66 2.52 2.37 2.29 2.2 2.11 2.02 1.92 1.8
60 7.08 4.98 4.19 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.5 2.35 2.2 2.12 2.03 1.94 1.84 1.73 1.6
120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.58 1.38
∞ 6.63 4.61 3.78 3.32 3.02 2.8 2.64 2.51 2.41 2.32 2.18 2.04 1.88 1.79 1.7 1.59 1.47 1.32 1
∂1 = degrees of freedom for numerator
∂2 = degrees of freedom for denominator

 

 

TABLE 4.3
Percentile Values for Student’s t Distribution [5]
dof(∂) t_{.60} t_{.70} t_{.80} t_{.90} t_{.95} t_{.975} t_{.99} t_{.995}
1 0.325 0.727 1.376 3.078 6.314 12.706 31.821 63.657
2 0.289 0.617 1.061 1.886 2.92 4.303 6.965 9.925
3 0.277 0.584 0.978 1.648 2.353 3.182 4.541 5.841
4 0.271 0.569 0.941 1.533 2.132 2.776 3.747 4.604
5 0.267 0.559 0.920 1.476 2.015 2.571 3.365 4.032
6 0.265 0.553 0.906 1.440 1.943 2.447 3.143 3.707
7 0.263 0.549 0.896 1.415 1.895 2.365 2.998 3.499
8 0.262 0.546 0.889 1.397 1.860 2.306 2.896 3.355
9 0.261 0.543 0.883 1.383 1.833 2.262 2.821 3.250
10 0.260 0.542 0.879 1.372 1.812 2.228 2.764 3.169
11 0.260 0.540 0.876 1.363 1.796 2.201 2.718 3.106
12 0.259 0.539 0.873 1.356 1.782 2.179 2.681 3.055
13 0.259 0.538 0.870 1.350 1.771 2.160 2.650 3.012
14 0.258 0.537 0.868 1.345 1.761 2.145 2.624 2.977
15 0.258 0.536 0.866 1.341 1.753 2.131 2.602 2.947
16 0.258 0.535 0.865 1.337 1.746 2.120 2.583 2.921
17 0.257 0.534 0.863 1.333 1.740 2.110 2.567 2.898
18 0.257 0.534 0.862 1.330 1.734 2.101 2.552 2.878
19 0.257 0.533 0.861 1.328 1.729 2.093 2.539 2.861
20 0.257 0.533 0.860 1.325 1.725 2.086 2.528 2.845
21 0.257 0.532 0.859 1.323 1.721 2.080 2.518 2.831
22 0.256 0.532 0.858 1.321 1.717 2.074 2.508 2.819
23 0.256 0.532 0.858 1.319 1.714 2.069 2.500 2.807
24 0.256 0.531 0.857 1.318 1.711 2.064 2.492 2.797
25 0.256 0.531 0.856 1.316 1.708 2.060 2.485 2.787
26 0.256 0.531 0.856 1.315 1.706 2.056 2.479 2.779
27 0.256 0.531 0.855 1.314 1.703 2.052 2.473 2.771
28 0.256 0.530 0.855 1.313 1.701 2.048 2.467 2.763
29 0.256 0.530 0.854 1.311 1.699 2.045 2.462 2.756
30 0.256 0.530 0.854 1.310 1.697 2.042 2.457 2.750
40 0.255 0.529 0.851 1.303 1.684 2.021 2.423 2.704
60 0.254 0.527 0.848 1.296 1.671 2.000 2.390 2.660
120 0.254 0.526 0.845 1.289 1.658 1.980 2.358 2.617
∞ 0.253 0.524 0.842 1.282 1.645 1.960 2.326 2.576
4.16.1
4.16.2

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