A random sample is drawn from a population with mean = 61 and standard deviation = 5.1. [You may find it useful to reference the z table.]

a. Is the sampling distribution of the sample mean with n = 19 and n = 38 normally distributed? (Round the standard error to 3 decimal places.)

b. Can you conclude that the sampling distribution of the sample mean is normally distributed for both sample sizes?

multiple choice 1

• Yes, both the sample means will have a normal distribution.
• No, both the sample means will not have a normal distribution.
• No, only the sample mean with n = 19 will have a normal distribution.
• No, only the sample mean with n = 38 will have a normal distribution. Correct

c. If the sampling distribution of the sample mean is normally distributed with n = 19, then calculate the probability that the sample mean falls between 61 and 63. (If appropriate, round final answer to 4 decimal places.)

multiple choice 2

• We cannot assume that the sampling distribution of the sample mean is normally distributed.
• We can assume that the sampling distribution of the sample mean is normally distributed and the probability that the sample mean falls between 61 and 63 is Incorrect

d. If the sampling distribution of the sample mean is normally distributed with n = 38, then calculate the probability that the sample mean falls between 61 and 63. (If appropriate, round final answer to 4 decimal places.)

multiple choice 3

• We cannot assume that the sampling distribution of the sample mean is normally distributed.
• We can assume that the sampling distribution of the sample mean is normally distributed and the probability that the sample mean falls between 61 and 63 is Correct

TABLE 1 Standard Normal Curve Areas Entries in this table provide cumulative probabilities, that is, the area P (ZS - 2) under the curve to the left of -z. For example, P(Z S -1.52) = 0.0643. -Z 0 Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0000 0.0000 0.0000 0.0000 -3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 -3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.7 0.0001 0.0001 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.6 -3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 -3. 0.0005 0.0004 0.0004 0.0004 0.0003 -3.3 0.0005 0.0005 0.0004 0.0004 0.0004 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 -3.2 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 -3.1 0.0012 0.0010 0.0013 0.0013 0.0011 0.0011 0.0010 -3.0 0.0013 0.0012 0.0011 0.0014 0.0014 -2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 -2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 0.0028 0.0027 0.0026 -2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0038 0.0037 0.0036 -2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0049 0.0048 -2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0069 0.0068 0.0066 0.0064 -2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 -2.3 0.0107 0.0104 0.0136 0.0132 0.0129 0.0125 0.0122 0.01 19 0.0116 0.0113 0.01 10 -2.2 0.0139 0.0143 -2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 -1. 0.0322 0.0301 0.0294 0.0359 0.0351 0.0344 0.0336 0.0329 0.0314 0.0307 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 0.0446 0.0436 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -1.6 0.0571 0.0559 -1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0681 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 0.0968 0.0951 0.1131 0. 1093 0.1003 0.0985 0.1151 0.1112 0.1075 0.1056 0.1038 0.1020 0.1170 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1401 0.1379 10 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 -0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 -0.7 0.2389 0.2358 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 0.2420 0.2327 0.2451 -0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 -0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 -0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 0.3594 0.3557 0.3520 0.3483 -0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.4207 0.4052 0.3936 0.3859 -0.2 0.4168 0.4129 0.4090 0.4013 0.3974 0.3897 -0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 -0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.0 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.1 0.5398 0.6026 0.6064 0.6103 0.6141 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.3 0.6879 0.4 0.6591 0.6700 0.6736 0.6844 0.6554 0.6628 0.6664 0.6772 0.6808 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.5 0.6915 0.6950 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 0.8485 0.8508 0.8554 0.8599 0.8621 1.0 0.8413 0.8438 0.8461 0.8531 0.8577 0.8810 0.8830 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.9015 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 0.9332 0.9345 0.9495 0.9505 0.9535 0.9545 0.9452 0.9463 0.9474 0.9484 0.9515 0.9525 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 20 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 0.9772 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.1 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.2 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.4 0.9918 0.9920 0.9952 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.6 0.9953 0.9955 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.7 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.8 0.9974 0.9975 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0.9990 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 0.9995 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9996 0.9997 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.4 0.9998 0.9998 0.9998 3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.6 0.9998 0.9998 3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.8 1.0000 39 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Source: Probabilities calculated with Excel