Introduction to Statistics in Psychology PSY 201 Professor Greg Francis Lecture 20 Hypothesis testing of the mean Do nuclear plants make you sick UP TO NOW UNKNOWN if we know of all the information we need only is hard to get From H0 X From the sample n The sample size Standard deviation of the population then we can calculate standard error X n and apply the techniques of last time UNKNOWN when is unknown we use our best guess the standard deviation of the sample s X X 2 n 1 then the estimated standard error of the mean for the sampling distribution is s sX n 4 for a population X 2 N which implies that we already know 2 so why would we want to test H0 about 2 3 UNDERLYING DISTRIBUTION t SCORES using s in place of affects the sampling distribution of the mean changes its shape away from a normal distribution especially for small n for large sample sizes the shape is very close to a normal distribution we can transform the sample means into t scores just like z scores t X sX distribution centered on zero standard deviation of one shape depends on sample size degrees of freedom called the Student s t distributions symmetrical bell shaped centered on the mean changes shape as sample size changes 5 6 DEGREES OF FREEDOM DEGREES OF FREEDOM DEGREES OF FREEDOM mathematical concept or suppose I want to know the degrees of freedom of deviation scores from the sample data set for the t distribution calculating degrees of freedom d f is easy number of observations less the number of restrictions placed on the observations e g if I know d f n 1 I can calculate X 25 and can calculate deviation scores of X1 X2 10 then knowing one of the X1 X2 values means that I can calculate the other one one degree of freedom among the scores with a sample of size n 21 23 24 27 30 4 2 1 2 5 there is a unique t distribution for each d f but if I only knew four of the deviation scores I could get the other one because I know xi 0 these deviation scores have four degrees of freedom 7 8 9 HYPOTHESIS TESTING TABLE TABLE we use the t distribution just like we used the normal distribution for hypothesis testing values for common values for different d f t distributions area under the curve indicates probability of those scores find the critical values with a table which lists the critical Integre Technical Publishing Co Inc Moore McCabe November 16 2007 1 29 p m moore page T 11 Tables In this table you look for the probability 2 in the upper Integre Technical Publishing Co Inc one tailed two tailed November 16 2007 1 29 p m page T 11 Tables T 11 Table entry for p and C is the critical value t with probability p lying to its right and probability C lying between t and t Probability p Table entry for p and C is the critical value t with probability p lying to its right and probability C lying between t and t t TABLE D Upper tail probability p df 25 20 15 10 05 025 02 01 005 0025 001 0005 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 80 100 1000 z 1 000 0 816 0 765 0 741 0 727 0 718 0 711 0 706 0 703 0 700 0 697 0 695 0 694 0 692 0 691 0 690 0 689 0 688 0 688 0 687 0 686 0 686 0 685 0 685 0 684 0 684 0 684 0 683 0 683 0 683 0 681 0 679 0 679 0 678 0 677 0 675 0 674 1 376 1 061 0 978 0 941 0 920 0 906 0 896 0 889 0 883 0 879 0 876 0 873 0 870 0 868 0 866 0 865 0 863 0 862 0 861 0 860 0 859 0 858 0 858 0 857 0 856 0 856 0 855 0 855 0 854 0 854 0 851 0 849 0 848 0 846 0 845 0 842 0 841 1 963 1 386 1 250 1 190 1 156 1 134 1 119 1 108 1 100 1 093 1 088 1 083 1 079 1 076 1 074 1 071 1 069 1 067 1 066 1 064 1 063 1 061 1 060 1 059 1 058 1 058 1 057 1 056 1 055 1 055 1 050 1 047 1 045 1 043 1 042 1 037 1 036 3 078 1 886 1 638 1 533 1 476 1 440 1 415 1 397 1 383 1 372 1 363 1 356 1 350 1 345 1 341 1 337 1 333 1 330 1 328 1 325 1 323 1 321 1 319 1 318 1 316 1 315 1 314 1 313 1 311 1 310 1 303 1 299 1 296 1 292 1 290 1 282 1 282 6 314 2 920 2 353 2 132 2 015 1 943 1 895 1 860 1 833 1 812 1 796 1 782 1 771 1 761 1 753 1 746 1 740 1 734 1 729 1 725 1 721 1 717 1 714 1 711 1 708 1 706 1 703 1 701 1 699 1 697 1 684 1 676 1 671 1 664 1 660 1 646 1 645 12 71 4 303 3 182 2 776 2 571 2 447 2 365 2 306 2 262 2 228 2 201 2 179 2 160 2 145 2 131 2 120 2 110 2 101 2 093 2 086 2 080 2 074 2 069 2 064 2 060 2 056 2 052 2 048 2 045 2 042 2 021 2 009 2 000 1 990 1 984 1 962 1 960 15 89 4 849 3 482 2 999 2 757 2 612 2 517 2 449 2 398 2 359 2 328 2 303 2 282 2 264 2 249 2 235 2 224 2 214 2 205 2 197 2 189 2 183 2 177 2 172 2 167 2 162 2 158 2 154 2 150 2 147 2 123 2 109 2 099 2 088 2 081 2 056 2 054 31 82 6 965 4 541 3 747 3 365 3 143 2 998 2 896 2 821 2 764 2 718 2 681 2 650 2 624 2 602 2 583 2 567 2 552 2 539 2 528 2 518 2 508 2 500 2 492 2 485 2 479 2 473 2 467 2 462 2 457 2 423 2 403 2 390 2 374 2 364 2 330 2 326 63 66 9 925 5 841 4 604 4 032 3 …