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by: Roxane Kuhn

24

0

5

# Linear Algebra & Probability MATH 202

Roxane Kuhn
USF
GPA 3.52

Peter Pacheco

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COURSE
PROF.
Peter Pacheco
TYPE
Class Notes
PAGES
5
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 5 page Class Notes was uploaded by Roxane Kuhn on Thursday October 29, 2015. The Class Notes belongs to MATH 202 at University of San Francisco taught by Peter Pacheco in Fall. Since its upload, it has received 24 views. For similar materials see /class/231227/math-202-university-of-san-francisco in Mathematics (M) at University of San Francisco.

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Date Created: 10/29/15
TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Exploring Data Distributions 0 Look for overall pattern shape7 center7 spread and deviations outliers 0 Mean use a calculator f z1z2xn71 71 TnZM Standard deviation use a calculator 0 Median Arrange all observations from smallest to largest The median M is located 71 12 observations from the beginning of this list 0 Quartiles The rst quartile Q1 is the median of the observations Whose position in the ordered list is to the left of the location of the overall median The third quartile Q3 is the median of the observations to the right of the location of the overall median o Five number summary Minimum7 Q1 M7 Q3 Maximum 0 Standardized value of m Exploring Data Relationships 0 Look for overall pattern form7 direction7 strength and deviations outliers7 in uential observations 0 Correlation use a calculator 0 Least squares regression line use a calculator 37 a bx with slope b rsysm and intercept a y 7 I o Residuals residual observed y 7 predicted y y 7 g Producing Data 0 Simple random sample Choose an SRS by giv ing every individual in the population a numer ical label and using Table B of random digits to choose the sample 0 Randomized comparative experiments Group 1 Treatment 1 Random Observe Allocation Res onse Group 2 Treatment 2 l p Probability and Sampling Distributions 0 Probability rules 0 Any probability satis es 0 PA 1 o The sample space S has probability PS 1 o For any event A7 PA does not occur 1 7 PA o If events A and B are disjoint7 PA or B PA PB 0 Sampling distribution of a sample mean 0 E has mean u and standard deviation a o E has a Normal distribution if the popula tion distribution is Normal 0 Central limit theorem 5 is approximately Normal when n is large Basics of Inference o 2 con dence interval for a population mean a known SRS from Normal population 51 51 2 from N0 1 0 Sample size for desired margin of error m gt2 n m o 2 test statistic for H0 u 120 a known SRS from Normal population 2 2 P values from N0 1 Inference About Means o 25 con dence interval for a population mean SRS from Normal population 8 5125 71 o ttest statistic for H0 u 120 SRS from Normal population 25 from tn 71 t T i 0 s 71 Matched pairs To compare the responses to the two treatments apply the one sample t proce dures to the observed differences P values from tn 7 1 o Two sample 25 con dence interval for 121 7 122 in dependent SRSs from Normal populations 2 2 S S E1752it1 1 2 711 712 with conservative 25 from t with df the smaller of m 7 1 and n2 7 1 or use software 0 Two sample t test statistic for H0 121 122 independent SRSs from Normal populations 1 2 75 2 2 2 n1 712 with conservative P values from t with df the smaller of m 7 1 and n2 7 1 or use software Inference About Proportions 0 Sampling distribution of a sample proportion when the population and the sample size are both large and 19 is not close to 0 or 1 13 is ap proximately Normal with mean 19 and standard deviation 19 1719 771 o Largesample 2 con dence interval for 19 231723 13 i 2 2 from N0 1 Plus four to greatly improve accuracy use the same formula after adding 2 successes and two failures to the data 0 2 test statistic for H0 19 190 large SRS 2 A P values from N0 1 1901 i 190 71 0 Sample size for desired margin of error m 2 2 lt5 mm where 19 is a guessed value for 19 or 19 05 o Largesample 2 con dence interval for 191 7 192 131 7 132 i 2SE 2 from N0 1 where the standard error of 131 7 132 is 1311 131 1321 132 n1 712 SE Plus four to greatly improve accuracy use the same formulas after adding one success and one failure to each sample o Two sample 2 test statistic for H0 p1 pg 0 tcon dence interval for regression slope 3 large independent SRSs b i tSEb 25 from tn 7 2 Z 131 i 132 AG A lt 1 1 gt o ttest statistic for no linear relationship p p m 72 H0 0 where 13 is the pooled proportion of successes t i PVa1ues from to 7 2 SE17 The ChiSquare Test 0 25 con dence interval for mean response My when 7 1 0 Expected count for a cell in a two way table m m 39 g 1 F813 25 from tn 7 2 row total x column total ex ected count p table total 0 t prediction interval for an individual observa o Chi square test statistic for testing whether the tiOh y Wheh m f3 row and column variables in an r x 0 table are unrelated expected cell counts not too small 397 i ESE tt from n 7 2 X2 2 observed count 7 expected count2 expected count Oneway Analysis of Variance with P values from the chi square distribution Comparing Several Means with df r 71 x 071 0 ANOVA F tests whether all of I populations have the same mean based on independent SRSs from I Normal populations with the same a P values come from the F distribution with I 7 1 and N 7 I degrees of freedom where N is the Describe the relationship using percents com parison of observed with expected counts and terms of X 2 Inference for Regression total observations in all samples 0 The regression model We have n observations 39 Describe the data heihg the I sample means and on x and y The response y for any xed m has standard deviations and sideby side graphs of a Normal distribution with mean given by the the samples true regression line My 04 1 x and standard deviation 0 Parameters are 04 B a The ANOVA F test statistic use software is F MSGMSE where 0 Estimate Oz by the intercept a and B by the slope b of the least squares line Estimate 0 by the MSG regression standard error n1517f2mfi7f2 I71 7 2 7 2 MSE 711 951 011 951 N 7 I 7 A 2 s 7 n 2 E residual Use software for all standard errors in regression Standard Normal probabilities z 00 01 02 03 04 05 06 07 08 09 734 0003 0003 0003 0003 0003 0003 0003 0003 0003 0002 733 0005 0005 0005 0004 0004 0004 0004 0004 0004 0003 732 0007 0007 0006 0006 0006 0006 0006 0005 0005 0005 731 0010 0009 0009 0009 0008 0008 0008 0008 0007 0007 730 0013 0013 0013 0012 0012 0011 0011 0011 0010 0010 729 0019 0018 0018 0017 0016 0016 0015 0015 0014 0014 728 0026 0025 0024 0023 0023 0022 0021 0021 0020 0019 727 0035 0034 0033 0032 0031 0030 0029 0028 0027 0026 726 0047 0045 0044 0043 0041 0040 0039 0038 0037 0036 725 0062 0060 0059 0057 0055 0054 0052 0051 0049 0048 724 0082 0080 0078 0075 0073 0071 0069 0068 0066 0064 723 0107 0104 0102 0099 0096 0094 0091 0089 0087 0084 722 0139 0136 0132 0129 0125 0122 0119 0116 0113 0110 721 0179 0174 0170 0166 0162 0158 0154 0150 0146 0143 720 0228 0222 0217 0212 0207 0202 0197 0192 0188 0183 719 0287 0281 0274 0268 0262 0256 0250 0244 0239 0233 718 0359 0351 0344 0336 0329 0322 0314 0307 0301 0294 717 0446 0436 0427 0418 0409 0401 0392 0384 0375 0367 716 0548 0537 0526 0516 0505 0495 0485 0475 0465 0455 715 0668 0655 0643 0630 0618 0606 0594 0582 0571 0559 714 0808 0793 0778 0764 0749 0735 0721 0708 0694 0681 713 0968 0951 0934 0918 0901 0885 0869 0853 0838 0823 712 1151 1131 1112 1093 1075 1056 1038 1020 1003 0985 711 1357 1335 1314 1292 1271 1251 1230 1210 1190 1170 710 1587 1562 1539 1515 1492 1469 1446 1423 1401 1379 709 1841 1814 1788 1762 1736 1711 1685 1660 1635 1611 708 2119 2090 2061 2033 2005 1977 1949 1922 1894 1867 707 2420 2389 2358 2327 2296 2266 2236 2206 2177 2148 706 2743 2709 2676 2643 2611 2578 2546 2514 2483 2451 705 3085 3050 3015 2981 2946 2912 2877 2843 2810 2776 704 3446 3409 3372 3336 3300 3264 3228 3192 3156 3121 703 3821 3783 3745 3707 3669 3632 3594 3557 3520 3483 702 4207 4168 4129 4090 4052 4013 3974 3936 3897 3859 701 4602 4562 4522 4483 4443 4404 4364 4325 4286 4247 700 5000 4960 4920 4880 4840 4801 4761 4721 4681 4641 00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359 01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753 02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141 03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517 04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879 05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224 06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549 07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852 08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133 09 8159 8186 8212 8238 8264 8289 8315 8340 8365 8389 10 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621 11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830 12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015 13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177 14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319 15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441 16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545 17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633 18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706 19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767 20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817 21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857 22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890 23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916 24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936 25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952 26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964 27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974 28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981 29 9981 9982 9982 9983 9984 9984 9985 9985 9986 9986 30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990 31 9990 9991 9991 9991 9992 9992 9992 9992 9993 9993 32 9993 9993 9994 9994 9994 9994 9994 9995 9995 9995 33 9995 9995 9995 9996 9996 9996 9996 9996 9996 9997 34 9997 9997 9997 9997 9997 9997 9997 9997 9997 9998 TABLE B Random digits Line 101 19223 95034 05756 28713 96409 12531 42544 82853 102 73676 47150 99400 01927 27754 42648 82425 36290 103 45467 71709 77558 00095 32863 29485 82226 90056 104 52711 38889 93074 60227 40011 85848 48767 52573 105 95592 94007 69971 91481 60779 53791 17297 59335 106 68417 35013 15529 72765 85089 57067 50211 47487 107 82739 57890 20807 47511 81676 55300 94383 14893 108 60940 72024 17868 24943 61790 90656 87964 18883 109 36009 19365 15412 39638 85453 46816 83485 41979 110 38448 48789 18338 24697 39364 42006 76688 08708 111 81486 69487 60513 09297 00412 71238 27649 39950 112 59636 88804 04634 71197 19352 73089 84898 45785 113 62568 70206 40325 03699 71080 22553 11486 11776 114 45149 32992 75730 66280 03819 56202 02938 70915 115 61041 77684 94322 24709 73698 14526 31893 32592 116 14459 26056 31424 80371 65103 62253 50490 61181 117 38167 98532 62183 70632 23417 26185 41448 75532 118 73190 32533 04470 29669 84407 90785 65956 86382 119 95857 07118 87664 92099 58806 66979 98624 84826 120 35476 55972 39421 65850 04266 35435 43742 11937 121 71487 09984 29077 14863 61683 47052 62224 51025 122 13873 81598 95052 90908 73592 75186 87136 95761 123 54580 81507 27102 56027 55892 33063 41842 81868 124 71035 09001 43367 49497 72719 96758 27611 91596 125 96746 12149 37823 71868 18442 35119 62103 39244 TABLE C t distribution critical values Upper tail probability 17 if 25 20 15 10 05 025 02 01 005 0025 001 0005 1 1000 1376 1963 3078 6314 1271 1589 3182 6366 1273 3183 6366 2 0816 1061 1386 1886 2920 4303 4849 6965 9925 1409 2233 3160 3 0765 0978 1250 1638 2353 3182 3482 4541 5841 7453 1021 1292 4 0741 0941 1190 1533 2132 2776 2999 3747 4604 5598 7173 8610 5 0727 0920 1156 1476 2015 2571 2757 3365 4032 4773 5893 6869 6 0718 0906 1134 1440 1943 2447 2612 3143 3707 4317 5208 5959 7 0711 0896 1119 1415 1895 2365 2517 2998 3499 4029 4785 5408 8 0706 0889 1108 1397 1860 2306 2449 2896 3355 3833 4501 5041 9 0 703 0883 1100 1383 1833 2262 2398 2821 3250 3690 4297 4781 10 0700 0879 1093 1372 1812 2228 2359 2764 3169 3581 4144 4587 11 0697 0876 1088 1363 1796 2201 2328 2718 3106 3497 4025 4437 12 0695 0873 1083 1356 1782 2179 2303 2681 3055 3428 3930 4318 13 0694 0870 1079 1350 1771 2160 2282 2650 3012 3372 3852 4221 14 0692 0868 1076 1345 1761 2145 2264 2624 2977 3326 3787 4140 15 0691 0866 1074 1341 1753 2131 2249 2602 2947 3286 3733 4073 16 0690 0865 1071 1337 1746 2120 2235 2583 2921 3252 3686 4015 17 0689 0863 1069 1333 1740 2110 2224 2567 2898 3222 3646 3965 18 0688 0862 1067 1330 1734 2101 2214 2552 2878 3197 3611 3922 19 0688 0861 1066 1328 1729 2093 2205 2539 2861 3174 3579 3883 20 0687 0860 1064 1325 1725 2086 2197 2528 2845 3153 3552 3850 21 0686 0859 1063 1323 1721 2080 2189 2518 2831 3135 3527 3819 22 0686 0858 1061 1321 1717 2074 2183 2508 2819 3119 3505 3792 23 0685 0858 1060 1319 1714 2069 2177 2500 2807 3104 3485 3768 24 0685 0857 1059 1318 1711 2064 2172 2492 2797 3091 3467 3745 25 0684 0856 1058 1316 1708 2060 2167 2485 2787 3078 3450 3725 26 0684 0856 1058 1315 1706 2056 2162 2479 2779 3067 3435 3707 27 0684 0855 1057 1314 1703 2052 2158 2473 2771 3057 3421 3690 28 0683 0855 1056 1313 1701 2048 2154 2467 2763 3047 3408 3674 29 0683 0854 1055 1311 1699 2045 2150 2462 2756 3038 3396 3659 30 0683 0854 1055 1310 1697 2042 2147 2457 2750 3030 3385 3646 40 0681 0851 1050 1303 1684 2021 2123 2423 2704 2971 3307 3551 50 0679 0849 1047 1299 1676 2009 2109 2403 2678 2937 3261 3496 60 0679 0848 1045 1296 1671 2000 2099 2390 2660 2915 3232 3460 80 0678 0846 1043 1292 1664 1990 2088 2374 2639 2887 3195 3416 100 0677 0845 1042 1290 1660 1984 2081 2364 2626 2871 3174 3390 1000 0675 0842 1037 1282 1646 1962 2056 2330 2581 2813 3098 3300 2 0674 0841 1036 1282 1645 1960 054 2326 2576 2807 3091 3291 50 60 70 80 90 95 96 98 99 995 998 999 on dence level C

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