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Standard Normal and t-Distributions: Critical Values, Exercises of Business Statistics

St Basils CollegeBusiness Statistics

Critical values for both the standard normal distribution and the t-distribution at various levels of confidence and degrees of freedom. It can be used as a reference for statistical hypothesis testing and confidence interval calculations.

What you will learn

  • What are the critical values for the standard normal distribution at a given level of confidence?
  • What is the significance of critical values in statistical hypothesis testing?
  • How do critical values for the t-distribution vary with degrees of freedom and confidence level?

Typology: Exercises

2021/2022

Uploaded on 09/12/2022

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Table
4-
Standard Normal Distribution
0
z
.09 .08
.07
.06
.OS
.04
.03 .02 .01
.00
-3.4
.0002 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003
-3.3
.0003 .0004 .0004 .0004 .0004 .0004 .0004 .0005 .0005 .0005
-3.2
.0005 .0005 .0005 .0006 .0006 .0006 .0006 .0006 .0007 .0007
-3.1
.0007 .0007 .0008 .0008 .0008 .0008 .0009 .0009 .0009 .0010
-3.0
.0010 .0010 .0011 .0011 .0011 .0012 .0012 .0013 .0013 .0013
-2.9
.0014 .0014 .0015 .0015 .0016 .0016 .0017 .0018 .0018 .0019
-2.8
.0019 .0020 .0021 .0021 .0022 .0023 .0023 .0024 .0025 .0026
-2.7
.0026 .0027 .0028 .0029 .0030 .0031 .0032 .0033 .0034 .0035
-2.6
.0036 .0037 .0038 .0039 .0040 .0041 .0043 .0044 .0045 .0047
-2.5
.0048 .0049 .0051 .0052 .0054 .0055 .0057 .0059 .0060 .0062
-2.4
.0064 .0066 .0068 .0069 .0071 .0073 .0075 .0078 .0080 .0082
-2.3
.0084 .0087 .0089 .0091 .0094 .0096 .0099 .0102 .0104 .0107
-2.2
.0110 .0113 .0116 .0119 .0122 .0125 .0129 .0132 .0136 .0139
-2.1
.0143 .0146 .0150 .0154 .0158 .0162 .0166 .0170 .0174 .0179
-2.0
.0183 .0188 .0192 .0197 .0202 .0207 .0212 .0217 .0222 .0228
-1.9
.0233 .0239 .0244 .0250 .0256 .0262 .0268 .0274 .0281 .0287
-1.8
.0294 .0301 .0307 .0314 .0322 .0329 .0336 .0344 .0351 .0359
-1.7
.0367 .0375 .0384 .0392 .0401 .0409 .0418 .0427 .0436 .0446
-1.6
.0455 .0465 .0475 .0485 .0495 .0505 .0516 .0526 .0537 .0548
-1.5
.0559 .0571 .0582 .0594 .0606 .0618 .0630 .0643 .0655 .0668
-1.4
.0681 .0694 .0708 .0721 .0735 .0749 .0764 .0778 .0793 .0808
-1.3
.0823 .0838 .0853 .0869 .0885 .0901 .0918 .0934 .0951 .0968
-1.2
.0985 .1003 .1020 .1038 .1056 .1075 .1093 .1112 .1131 .1151
-1.1
.1170 .1190 .1210 .1230 .1251 .1271 .1292 .1314 .1335 .1357
-1.0
.1379 .1401 .1423 .1446 .1469 .1492 .1515 .1539 .1562 .1587
-0.9
.1611 .1635 .1660 .1685 .1711 .1736 .1762 .1788 .1814 .1841
-0.8
.1867 .1894 .1922 .1949 .1977 .2005 .2033 .2061 .2090 .2119
-0.7
.2148 .2177 .2206 .2236 .2266 .2296 .2327 .2358 .2389 .2420
-0.6
.2451 .2483 .2514 .2546 .2578 .2611 .2643 .2676 .2709 .2743
-o.s .2776 .2810 .2843 .2877 .2912 .2946 .2981 .3015 .3050 .3085
-0.4
.3121 .3156 .3192 .3228 .3264 .3300 .3336 .3372 .3409 .3446
-0.3
.3483 .3520 .3557 .3594 .3632 .3669 .3707 .3745 .3783 .3821
-0.2
.3859 .3897 .3936 .3974 .4013 .4052 .4090 .4129 .4168 .4207
-0.1
.4247 .4286 .4325 .4364 .4404 .4443 .4483 .4521 .4562 .4602
-0.0
.4641 .4681 .4721 .4761 .4801 .4840 .4880 .4920 .4960 .5000
Critical Values
Level
of
Confidence c
z,
0.80
1.28
0.90
1.645
0.95
1.96
0.99 2.575
-z,
z=O
z,
pf3
pf4

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Table 4- Standard Normal Distribution

Critical Values

Table 4- Standard Normal Distribution (continued)

Table 6- Chi-Square Distribution Right tail Two tails

 - z .09 .08 .07 .06 .OS .04 .03 .02 .01. - -3.4 .0002 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003. - -3.3 .0003 .0004 .0004 .0004 .0004 .0004 .0004 .0005 .0005. - -3.2 .0005 .0005 .0005 .0006 .0006 .0006 .0006 .0006 .0007. - -3.1 .0007 .0007 .0008 .0008 .0008 .0008 .0009 .0009 .0009.
  • -3.0 .0010 .0010 .0011 .0011 .0011 .0012 .0012 .0013 .0013.
  • -2.9 .0014 .0014 .0015 .0015 .0016 .0016 .0017 .0018 .0018.
    • -2.8 .0019 .0020 .0021 .0021 .0022 .0023 .0023 .0024 .0025.
  • -2.7 .0026 .0027 .0028 .0029 .0030 .0031 .0032 .0033 .0034.
  • -2.6 .0036 .0037 .0038 .0039 .0040 .0041 .0043 .0044 .0045.
  • -2.5 .0048 .0049 .0051 .0052 .0054 .0055 .0057 .0059 .0060.
  • -2.4 .0064 .0066 .0068 .0069 .0071 .0073 .0075 .0078 .0080.
  • -2.3 .0084 .0087 .0089 .0091 .0094 .0096 .0099 .0102 .0104.
  • -2.2 .0110 .0113 .0116 .0119 .0122 .0125 .0129 .0132 .0136.
  • -2.1 .0143 .0146 .0150 .0154 .0158 .0162 .0166 .0170 .0174.
  • -2.0 .0183 .0188 .0192 .0197 .0202 .0207 .0212 .0217 .0222.
  • -1.9 .0233 .0239 .0244 .0250 .0256 .0262 .0268 .0274 .0281.
  • -1.8 .0294 .0301 .0307 .0314 .0322 .0329 .0336 .0344 .0351.
  • -1.7 .0367 .0375 .0384 .0392 .0401 .0409 .0418 .0427 .0436.
  • -1.6 .0455 .0465 .0475 .0485 .0495 .0505 .0516 .0526 .0537.
  • -1.5 .0559 .0571 .0582 .0594 .0606 .0618 .0630 .0643 .0655.
    • -1.4 .0681 .0694 .0708 .0721 .0735 .0749 .0764 .0778 .0793.
  • -1.3 .0823 .0838 .0853 .0869 .0885 .0901 .0918 .0934 .0951.
    • -1.2 .0985 .1003 .1020 .1038 .1056 .1075 .1093 .1112 .1131.
    • -1.1 .1170 .1190 .1210 .1230 .1251 .1271 .1292 .1314 .1335.
    • -1.0 .1379 .1401 .1423 .1446 .1469 .1492 .1515 .1539 .1562.
    • -0.9 .1611 .1635 .1660 .1685 .1711 .1736 .1762 .1788 .1814.
  • -0.8 .1867 .1894 .1922 .1949 .1977 .2005 .2033 .2061 .2090.
  • -0.7 .2148 .2177 .2206 .2236 .2266 .2296 .2327 .2358 .2389.
  • -0.6 .2451 .2483 .2514 .2546 .2578 .2611 .2643 .2676 .2709.
  • -o.s .2776 .2810 .2843 .2877 .2912 .2946 .2981 .3015 .3050.
  • -0.4 .3121 .3156 .3192 .3228 .3264 .3300 .3336 .3372 .3409.
  • -0.3 .3483 .3520 .3557 .3594 .3632 .3669 .3707 .3745 .3783.
  • -0.2 .3859 .3897 .3936 .3974 .4013 .4052 .4090 .4129 .4168.
  • -0.1 .4247 .4286 .4325 .4364 .4404 .4443 .4483 .4521 .4562.
  • -0.0 .4641 .4681 .4721 .4761 .4801 .4840 .4880 .4920 .4960. - 0.80 1. Level of Confidence c z, - 0.90 1. - 0.95 1. - z .00 .01 .02 .03 .04 .OS .06 .07 .08. - 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319.
    • 0.1 .5398 .5438 .5478 .55J7 .5557 .5596 .5636 .5675 .5714.
      • 0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103.
    • 0.3 .6179 .6217 .6255 .6293 .6331 .6368' .•^6400 .6443 .6480.
      • 0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844.
    • 0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190.
      • 0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517.
      • 0.7 .758{) .7611 .7642 .7673 .7704 ;7734 .7764 .7794 .7823.
      • 0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106.
    • 0.9 .8159 .8186 .8212 .8238 .8264 .8289 ;8315 .8340 .8365.
      • 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599.
    • 1.1 .8643 .8665 .8686 .8708 .8729 .8749 ~8770 .8790 .8810.
      • 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997.
    • 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162.
    • 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306.
    • 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429.
      • 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535.
    • 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625.
    • 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699.
    • 1.9 .9713 ,9719 .9726 .9732 ;9738 .9744 .9750 .9756 .9761.
    • 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812.
    • 2.1 .9821 .9826 .9830 .9834 ;9838 .9842 .9846 .9850 .9854.
    • 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887.
  • 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913.
    • 2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934.
    • 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951.
    • 2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963.
    • 2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973.
    • 2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980.
    • 2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986.
    • 3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990.
  • 3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993.
  • 3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995.
  • 3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996.
  • 3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997. - confidence, c 0.80 0.90 0.95 0.98 0. Level of - One tail, a 0.10 0.05 0.025 0.01 0. - d.f. Two tails, a 0.20 0.10 0.05 0.02 0. - 2 1.886 2.920 4.303 6.965 9. 1 3.078 6.314 12.706 31.821 63.657 -t - 4 1.533 2.132 2.776 3.747 4. 3 1.638 2.353 3.182 4.541 5.841 c-confidence interval - 5 1.476 2.015 2.571 3.365 4. - 6 1.440 1.943 2.447 3.143 3. - 7 1.415 1.895 2.365 2.998 3. - 8 1.397 1.860 2'.306 2.896 3. - 9 1.383 1.833 2.262 2.821 3. - 10 1.372 1.812 2.228 2.764 3. - 11 1.363 1.796 2.201 2.718 3. - 13 1.350 1.771 2.160 2.650 3. 12 1.356 1.782 2.179 2.681 3.055 Left-tailed test - 14 1.345 1.761 2.145 2.624 2. - 15 1.341 1.753 2.131 2.602 2. - 16 1.337 1.746 2.120 2.583 2. - 17 1.333 1.740 2.110 2.567 2. - 18 1.330 1.734 2.101 2.552 2. - 19 1.328 1.729 2.093 2.539 2. - 20 1.325 1.725 2.086 2.528 2. - 22 1.321 1.117 2.074 2.508 2. 21 1.323 1.721 2.080 2.518 2.831 Right-tailed test - 23 1.319 1.714 2.069 2.500 2. - 24 1.318 1.711 2.064 2.492 2. - 25 1.316 1.708 2.060 2.485 2. - 26 1.315 1.706 2.056 2.479 2. - 27 1.314 1.703 2.052 2.473 2. - 29 1.311 1.699 2.045 2.462 2. 28 1.313 1.701 2.048 2.467 2.763 -t - 31 1.309 1.696 2.040 2.453 2. 30 1.310 1.697 2.{)42 2.457 2.750 Two-tailed test - 32 1.309 1.694 2.{)37 2.449 2. - 33 1.308 1.692 2.035 2.445 2. - 34 1.307 1.691 2.032 2.441 2. - 35 1.306 1.690 2.030 2.438 2. - 36 1.306 1.688 2.028 2.434 2. - 37 1.305 1.687 2.026 2.431 2. - 38 1.304 1.686 2.024 2.429 2. - 39 1.304 1.685 2.023 2.426 2. - 40 1.303 1.684 2.021 2.423 2. - 45 1.301 1.679 2.014 2.412 2. - so 1.299 1.676 2.009 2.403 2. - 60 1.296 1.671 2.000 2.390 2. - 70 1.294 1.667 1.994 2.381 2. - 80 1.292 1.664 1.990 2.374 2. - 90 1.291 1.662 1.987 2.368 2. - 100 1.290 1.660 1.984 2.364 2.
    • 500 1.283 1.648 1.965 2.334 2.
  • 1000 1.282 1.646 1.962 2.330 2. - 00 1.282 1.645 1.960 2.326 2.
  • freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0. Degrees of a - 1 - - 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7. - 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10. - 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12. - 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14. - 5 0.412 0.554 0.831 1.145 1.610 9.236 11.071 12.833 15.086 16. - 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18. - 7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20. - 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21. - 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23. - 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25. - 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26. - 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28. - 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29. - 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31. - 15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32. - 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34. - 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35. - 18 6.265 7.015 8.;m 9.390 10.865 25.989 28.869 31.526 34.805 37. - 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38. - 20 7.434 8.260 9.591 10.851 12.443 28.412 31+410 34.170 37.566 39. - 21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41. - 22 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42. - 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44. - 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45. - 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46. - 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48. - 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49. - 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50. - 29 13.121 14.257 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52. - 30 13.787 14.954 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53. - 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66. - so 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79. - 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91. - 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104. - 80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116. - 90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.
    • 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.