9.12 Independent random samples selected from two normal populations produced the following sample means and standard deviations: Sample 1 Sample 2 n = = 17 x = 5.4 $1 = 3.4 n2 = 12 2 = 7.9 = 82 = 4.8 a. Assuming equal variances, conduct the test Ho: ( - 2) = 0, Ha (12) 0 using a = .05. b. Find and interpret the 95% confidence interval for (1 - 2). against A study of the differences in cognitive function between normal individuals and patients diagnosed with schizophrenia was published in the American Journal of Psychiatry (Apr. 2010). The total time (in minutes) a subject spent on the Trail Making Test (a standard psychological test) was used as a measure of cognitive function. The researchers theorize that the mean time on the Trail Making Test for schizophrenics will be larger than the corresponding mean for normal subjects. The data for independent random samples of 41 schizophrenics and 49 normal individuals yielded the following results: Schizophrenia Normal Sample size 41 49 Mean time 104.23 62.24 Standard deviation 45.45 16.34 a. Define the parameter of interest to the researchers. b. Set up the null and alternative hypothesis for testing the researchers' theory. c. The researchers conducted the test, part b, and reported a p-value of .001. What conclusions can you draw from this result? (Use a = .01.) d. Find a 99% confidence interval for the target parameter. Interpret the result. Does your conclusion agree with that of part c? TABLE 1 (continued) Z .00 .07 .08 .09 .01 .02 .03 .04 .05 .06 .0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 .1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 .2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 .3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 .4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 .7224 .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 .9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 2.0 9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 2.4 .9918 .9920 .9922 ..9925 .9927 .9929 .9931 .9932 .9934 .9936 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964 2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990 3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 9993 3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995 3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997 3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998 Z Area 3.50 .99976737 4.00 .99996833 4.50 .99999660 5.00 .99999971 8 1.0 Table IV Critical Values of x f(x) a 0 Degrees of Freedom 2 2 2 X.995 X.990 2 X.975 2 X.950 X.900 10 1234567899 .0000393 .0001571 .0009821 .0039321 .0100251 .0201007 .0157908 .0506356 .102587 .0717212 .210720 .114832 .215795 .351846 .206990 .584375 .297110 .484419 .710721 .411740 1.063623 .554300 .831211 1.145476 1.61031 .675727 .872085 1.237347 1.63539 2.20413 .989265 1.239043 1.68987 2.16735 2.83311 1.344419 1.646482 2.17973 2.73264 3.48954 1.734926 2.087912 2.70039 3.32511 4.16816 2.15585 2.55821 3.24697 3.94030 4.86518 11 2.60321 3.05347 3.81575 4.57481 5.57779 12 3.07382 3.57056 4.40379 5.22603 6.30380 13 3.56503 4.10691 5.00874 5.89186 7.04150 14 4.07468 4.66043 5.62872 6.57063 7.78953 15 4.60094 5.22935 6.26214 7.26094 8.54675 16 5.14224 5.81221 6.90766 7.96164 9.31223 17 5.69724 6.40776 7.56418 8.67176 10.0852 18 6.26481 7.01491 8.23075 9.39046 10.8649 19 6.84398 7.63273 8.90655 10.1170 11.6509 20 7.43386 8.26040 9.59083 10.8508 12.4426 21 8.03366 8.89720 10.28293 11.5913 13.2396 90 22222222EERE 8.64272 9.54249 10.9823 12.3380 14.0415 9.26042 10.19567 11.6885 13.0905 14.8479 9.88623 10.8564 12.4011 13.8484 15.6587 10.5197 11.5240 13.1197 14.6114 16.4734 11.1603 12.1981 13.8439 15.3791 17.2919 11.8076 12.8786 14.5733 16.1513 18.1138 12.4613 13.5648 15.3079 16.9279 18.9392 13.1211 14.2565 16.0471 17.7083 19.7677 13.7867 14.9535 16.7908 18.4926 20.5992 40 20.7065 22.1643 24.4331 26.5093 29.0505 50 27.9907 29.7067 32.3574 34.7642 37.6886 60 35.5346 37.4848 40.4817 43.1879 46.4589 70 43.2752 45.4418 48.7576 51.7393 55.3290 64.2778 80 51.1720 53.5400 57.1532 60.3915 73.2912 59.1963 61.7541 65.6466 69.1260 82.3581 100 67.3276 70.0648 74.2219 77.9295 (continued)