If, in a sample of n = 20 selected from a normal population, X= 53 and S = 12, what are the critical values of t if the level of significance, a, is 0.01, the null hypothesis, Ho, is μ = 50, and the alternative hypothesis, H₁, is µ #50? Click here to view page 1 of the critical values for the t Distribution. Click here to view page 2 of the critical values for the t Distribution.

Critical Values of t. For a particular number of degrees of freedom, entry represents the critical value of t corresponding to the cumulative probability 1 minus alpha and a specified upper-tail area alpha. Cumulative Probabilities 0.75 0.90 0.95 0.975 0.99 0.995 Upper-Tail Areas Degrees of Freedom 0.25 0.10 0.05 0.025 0.01 0.005 1 1.0000 3.0777 6.3138 12.7062 31.8207 63.6574 2 0.8165 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.7649 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.7407 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.7267 1.4759 2.0150 2.5706 3.3649 4.0322 6 0.7176 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.7111 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.7064 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.7027 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.6998 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.6974 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.6955 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.6938 1.3502 1.7709 2.1604 2.6503 3.0123 14 0.6924 1.3450 1.7613 2.1448 2.6245 2.9768 15 0.6912 1.3406 1.7531 2.1315 2.6025 2.9467 16 0.6901 1.3368 1.7459 2.1199 2.5835 2.9208 17 0.6892 1.3334 1.7396 2.1098 2.5669 2.8982 18 0.6884 1.3304 1.7341 2.1009 2.5524 2.8784 19 0.6876 1.3277 1.7291 2.0930 2.5395 2.8609 20 0.6870 1.3253 1.7247 2.0860 2.5280 2.8453 21 0.6864 1.3232 1.7207 2.0796 2.5177 2.8314 22 0.6858 1.3212 1.7171 2.0739 2.5083 2.8188 23 0.6853 1.3195 1.7139 2.0687 2.4999 2.8073 24 0.6848 1.3178 1.7109 2.0639 2.4922 2.7969 25 0.6844 1.3163 1.7081 2.0595 2.4851 2.7874 26 0.6840 1.3150 1.7056 2.0555 2.4786 2.7787 27 0.6837 1.3137 1.7033 2.0518 2.4727 2.7707 28 0.6834 1.3125 1.7011 2.0484 2.4671 2.7633 29 0.6830 1.3114 1.6991 2.0452 2.4620 2.7564 30 0.6828 1.3104 1.6973 2.0423 2.4573 2.7500 31 0.6825 1.3095 1.6955 2.0395 2.4528 2.7440 32 0.6822 1.3086 1.6939 2.0369 2.4487 2.7385 33 0.6820 1.3077 1.6924 2.0345 2.4448 2.7333 34 0.6818 1.3070 1.6909 2.0322 2.4411 2.7284 35 0.6816 1.3062 1.6896 2.0301 2.4377 2.7238 36 0.6814 1.3055 1.6883 2.0281 2.4345 2.7195 37 0.6812 1.3049 1.6871 2.0262 2.4314 2.7154 38 0.6810 1.3042 1.6860 2.0244 2.4286 2.7116 39 0.6808 1.3036 1.6849 2.0227 2.4258 2.7079 40 0.6807 1.3031 1.6839 2.0211 2.4233 2.7045 41 0.6805 1.3025 1.6829 2.0195 2.4208 2.7012 42 0.6804 1.3020 1.6820 2.0181 2.4185 2.6981 43 0.6802 1.3016 1.6811 2.0167 2.4163 2.6951 44 0.6801 1.3011 1.6802 2.0154 2.4141 2.6923 45 0.6800 1.3006 1.6794 2.0141 2.4121 2.6896 46 0.6799 1.3002 1.6787 2.0129 2.4102 2.6870 47 0.6797 1.2998 1.6779 2.0117 2.4083 2.6846 48 0.6796 1.2994 1.6772 2.0106 2.4066 2.6822 Critical Values of t. Cumulative Probabilities 0.75 0.90 0.95 0.975 0.99 0.995 Upper-Tail Areas Degrees of Freedom 0.25 0.10 0.05 0.025 0.01 0.005 49 0.6795 1.2991 1.6766 2.0096 2.4049 2.6800 50 0.6794 1.2987 1.6759 2.0086 2.4033 2.6778 51 0.6793 1.2984 1.6753 2.0076 2.4017 2.6757 52 0.6792 1.2980 1.6747 2.0066 2.4002 2.6737 53 0.6791 1.2977 1.6741 2.0057 2.3988 2.6718 54 0.6791 1.2974 1.6736 2.0049 2.3974 2.6700 55 0.6790 1.2971 1.6730 2.0040 2.3961 2.6682 56 0.6789 1.2969 1.6725 2.0032 2.3948 2.6665 57 0.6788 1.2966 1.6720 2.0025 2.3936 2.6649 58 0.6787 1.2963 1.6716 2.0017 2.3924 2.6633 59 0.6787 1.2961 1.6711 2.0010 2.3912 2.6618 60 0.6786 1.2958 1.6706 2.0003 2.3901 2.6603 61 0.6785 1.2956 1.6702 1.9996 2.3890 2.6589 62 0.6785 1.2954 1.6698 1.9990 2.3880 2.6575 63 0.6784 1.2951 1.6694 1.9983 2.3870 2.6561 64 0.6783 1.2949 1.6690 1.9977 2.3860 2.6549 65 0.6783 1.2947 1.6686 1.9971 2.3851 2.6536 66 0.6782 1.2945 1.6683 1.9966 2.3842 2.6524 67 0.6782 1.2943 1.6679 1.9960 2.3833 2.6512 68 0.6781 1.2941 1.6676 1.9955 2.3824 2.6501 69 0.6781 1.2939 1.6672 1.9949 2.3816 2.6490 70 0.6780 1.2938 1.6669 1.9944 2.3808 2.6479 71 0.6780 1.2936 1.6666 1.9939 2.3800 2.6469 72 0.6779 1.2934 1.6663 1.9935 2.3793 2.6459 73 0.6779 1.2933 1.6660 1.9930 2.3785 2.6449 74 0.6778 1.2931 1.6657 1.9925 2.3778 2.6439 75 0.6778 1.2929 1.6654 1.9921 2.3771 2.6430 76 0.6777 1.2928 1.6652 1.9917 2.3764 2.6421 77 0.6777 1.2926 1.6649 1.9913 2.3758 2.6412 78 0.6776 1.2925 1.6646 1.9908 2.3751 2.6403 79 0.6776 1.2924 1.6644 1.9905 2.3745 2.6395 80 0.6776 1.2922 1.6641 1.9901 2.3739 2.6387 81 0.6775 1.2921 1.6639 1.9897 2.3733 2.6379 82 0.6775 1.2920 1.6636 1.9893 2.3727 2.6371 83 0.6775 1.2918 1.6634 1.9890 2.3721 2.6364 84 0.6774 1.2917 1.6632 1.9886 2.3716 2.6356 85 0.6774 1.2916 1.6630 1.9883 2.3710 2.6349 86 0.6774 1.2915 1.6628 1.9879 2.3705 2.6342 87 0.6773 1.2914 1.6626 1.9876 2.3700 2.6335 88 0.6773 1.2912 1.6624 1.9873 2.3695 2.6329 89 0.6773 1.2911 1.6622 1.9870 2.3690 2.6322 90 0.6772 1.2910 1.6620 1.9867 2.3685 2.6316 91 0.6772 1.2909 1.6618 1.9864 2.3680 2.6309 92 0.6772 1.2908 1.6616 1.9861 2.3676 2.6303 93 0.6771 1.2907 1.6614 1.9858 2.3671 2.6297 94 0.6771 1.2906 1.6612 1.9855 2.3667 2.6291 95 0.6771 1.2905 1.6611 1.9853 2.3662 2.6286 96 0.6771 1.2904 1.6609 1.9850 2.3658 2.6280 97 0.6770 1.2903 1.6607 1.9847 2.3654 2.6275 98 0.6770 1.2902 1.6606 1.9845 2.3650 2.6269 99 0.6770 1.2902 1.6604 1.9842 2.3646 2.6264 100 0.6770 1.2901 1.6602 1.9840 2.3642 2.6259 110 0.6767 1.2893 1.6588 1.9818 2.3607 2.6213 120 0.6765 1.2886 1.6577 1.9799 2.3578 2.6174 infinity 0.6745 1.2816 1.6449 1.9600 2.3263 2.5758