Question

6.57 Working on summer vacation. According to an Adweek/ Harris (July 2011) poll of U.S. adults, 35% do not work during their summer vacation. Assume that the true pro- portion of all U.S. adults that do not work during summer vacation is p = .35. Now consider a random sample of 500 U.S. adults.

Answer 1

a) Mean is calculated as

n*p

=500*0.35

=175

p(1-p) n

$=\sqrt{\frac{0.35(1-0.35)}{500}}=0.0213$

b) Since the sample is large enough to consider it as normal distribution hence probability is calculated using Z calculation as:

$Z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}=\frac{0.70-0.35}{0.0213}=16.432$

P($\hat{p}$>0.70) =P(Z>16.432)=0.000 computed using Z table shown below

c) Now P($\hat{p}$ >P_o )=0.01 we need to find Z score at P_o    using Z table shown below:

Z=2.33 so using Z formula as:

$Z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\Rightarrow 2.33=\frac{p_{o}-0.35}{0.0213}\Rightarrow p_{o}=0.40$

Table of Standard Normal Probabilities for Negative Z-scores Table of Standard Normal Probabilities for Positive Z-scores 0.00 0.5000 05398 0.03 0.5120 0.04 0.09 0.5359 0,00 0,06 0.5239 0.02 0.08 001 003 D.04 0,05 0.06 0.07 0,08 0.09 001 0.02 0,05 0.07 7 0,0002 0,0003 .0004 0.0006 0,0003 00 0.5040 0.5080 0.5199 0.5279 D.5675 0.6064 0.5319 -34 0,0003 0.0003 0.0003 0.0003 0.0003 00003 0.0003 0.5160 0.0005 0.0007 0.5438 0.5753 -3.3 0.0005 0.0005 0,0004 0,0004 00004 0.0004 0.0O04 0.0005 0.0007 0.0010 0.0014 D.D003 0.1 0.5478 0.5517 0.5557 0.5596 D5636 0.5714 -3.2 0.0007 0.0006 0.0009 0.0006 0,0006 00006 0.0005 0,0005 0,2 0.5793 0.5832 0.5871 0,5910 0.5948 0.587 0.6026 0.6103 0,6141 -3.1 0.0010 0.6517 0,6879 0.7224 D.DO09 a.0009 0.0008 0.0008 0.0008 00008 D.DO07 D.0010 0.0014 0.3 0.6179 0.6217 0.6255 0.6293 0,633 0.6368 D.6406 D.6443 0.6480 0,6554 0.6915 O0011 0.0015 06591 0,6950 -3.0 0.0013 0.D013 0.0013 0,0018 .0012 0.0012 0.0016 0.0011 0.0011 0.4 0.6628 0.6664 0.7019 0.6700 0.6736 0.7088 06772 0.7123 D.6808 0.6844 -2.9 0.0019 0,0018 0.0017 0.0016 0.0015 0,5 0,6985 07054 0.7157 0.7190 0.7517 D.0025 0.0034 0.0045 O7389 0.7704 0.7995 D.7486 0.7794 0.8078 DS340 D.8577 D8790 2.8 0.0026 0.0024 a.0023 0,0023 0.0031 0,0041 0.0022 0.0030 0,0040 O0021 0.0021 0.0020 D.DO19 0.6 0.7257 0.729 0.7324 0.7357 07422 D.7454 0.7549 0.7852 0.8133 0.7823 0.8106 -2.7 -2.6 0.0035 0 0017 0.0033 0,0044 0.0032 0.0029 00039 0.0028 0.0038 0.005 0.0027 0.0026 0.0036 0.7 0.8 0.7580 0.7881 0.7611 0.7910 0.7642 0.7673 0,7967 0.7734 0.8023 0.7764 0.805 0.0043 0.0037 0.7930 1 08389 0.8621 0.8159 0.8413 -25 0.0062 D.DO60 0.0059 a.0057 0.0055 0.0073 0.0054 00052 0.0049 D.D048 0.9 0.8186 0.8212 0,8238 .8264 0.8289 D8315 0.8365 0.8599 0.8810 0.0080 -24 0.0082 0,0078 0.0102 0.0075 0.0071 00059 0.0068 0.0066 0.0064 1.0 0.8438 a5665 0.8869 0.9049 0,8461 0.8485 0.8508 0.8531 0.3749 0.8554 03770 0.0107 0.0139 0.0179 0.009 0.8643 23 0.0104 a.0099 0.0096 0,0004 0.0122 0.0158 0.0089 0.0087 D.0O84 0.8686 0.8708 O.8725 0.8830 0.0136 0.0174 -2.2 -2.1 0.0132 0.0170 0.0129 0,0166 0,0125 0.0162 0.0110 0.0143 1.2 1.3 0.8907 0,9082 O.8925 0.8944 0.9115 0.8997 0,9015 00119 0.0116 0.0113 0.8849 0.8888 0.8962 0.913 09279 0.8980 00154 0.0150 0.0146 0.9032 0.9066 0.9147 0.9162 0.9177 09099 0,9222 0,9357 0,9474 0.9573 0.9656 0.9207 0,9236 0.9370 -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 00250 00314 0.0192 0.0188 0.0183 1.4 0.9192 0.9332 09251 0.9265 D.9292 0.9306 0.9319 0.9429 0.9535 0.9625 0.0287 0.0359 0.0256 0.0239 0.0233 1.5 0.9382 D.9418 D.9525 0,9441 -1.9 0.0281 0.0274 0,0268 0,0262 0.0244 0.9345 0.9394 0.9505 0.9599 0.9406 0.00 0.0329 0.0294 D.067 -L.8 0.035 0.0436 0.0344 0.0336 0.0322 0.007 1.6 0,9452 0.9554 0.9463 0,9484 09495 0.9515 0,9545 0.040 00495 0.0375 0,0465 0.057 0.9582 -1.7 0.0446 0.0427 0,0418 0,0409 00392 0.0384 1.7 0.9564 09591 0.9608 0.9616 0,9633 0048s 00594 0.0455 0.0559 09686 0.9750 0.0548 0.0526 0.0516 0.0475 0,9649 0.9706 16 0.0537 00505 1.8 0.9641 0,9664 09671 0.9678 0.9693 0.9699 0.9761 0.9812 0.9854 0.0668 0.0655 0,0643 0.0630 0.0618 0,0606 0.0582 1,9 09713 0,9719 0.9726 0,9732 09738 09744 0.9756 0,9767 -1.5 O0721 00869 0.1038 0.0708 0.0853 0.0808 D.U793 U.U778 14 .0764 .0749 0.0735 0.0694 D.0681 2,0 0,9772 0.9778 0.9783 0.9788 0.9793 09798 0.9803 D.98U8 0.9817 0.0968 0.1151 01357 0.1587 0.184 0.9834 0,987 -1.3 0.0918 0.0823 0.9826 09838 0.9857 0.0951 0.0934 0.0901 0.0885 0.0838 2.1 0.9821 0.9830 0.9842 0,9878 0.9846 D.9850 0.1020 0.1210 0988 0,9864 0,9868 -1.2 0.1131 0.1112 0,1093 0.1075 0.1056 0.125 0.1003 D.0985 2.2 0.9861 0.9875 0.9884 0.9887 0,9890 0.9901 0.9925 0.9906 0.9929 -1.1 0.1335 01314 .1292 0.1271 0.1492 O1230 0.1190 D.1170 23 0.9893 0,9806 0.9898 09904 0.9927 0.9909 0.9911 0.9913 0,9916 0.140 0993 0.9922 -10 0.1562 0.1539 0.1515 0.1469 0.1446 0.1423 0.1379 2.4 0.9918 0.9920 0.9940 09935 0.9932 0.9934 0.995 0.9936 0.1814 0.9946 -09 0.1788 a.1762 .1736 0.1711 0.1977 0.2266 O1685 0.1660 D.1635 D.1611 D.1867 2.5 0.9938 0.9941 0,9943 09945 0.9948 0.996 0.9971 D.9949 0.9952 0.2061 0.2358 0.9957 0,9968 0.9953 0.9965 D.9962 O.8 0.2119 0.2090 0.2033 0.2327 0.2005 0.2296 O.1949 0.1922 0.1894 2.6 0,9956 09959 0.9960 0.9963 0.9964 0.9967 0.9972 09973 D.9979 0.9985 0.9989 -07 0.2420 0.2389 02236 0.2206 0.2177 0.2148 D.245 0.2776 0.3121 2.7 0,9966 09969 0.9970 0,9974 0.2578 0.2912 0.3264 06 0.2743 0.2709 0.2676 d.2643 0.2611 02546 02514 0.2483 2.8 0.9974 0.9975 0.9976 0.9977 09977 0.9978 D.9979 0.9980 0.998 0.9984 0.9989 0.9981 0.9987 0,9982 0,9987 0.9985 0.9989 -05 -04 0.3085 0.34-16 0.3050 0.3409 D.3783 0.3015 0.3372 0,2981 0,3336 0.2946 0.3300 02877 03228 0.2843 0.3192 0.2810 0.3156 2,9 3.0 0.9982 0.9987 0,999 0.9983 0,9988 09984 0,9988 0.9986 0.9990 0.9986 0,9990 03 0.3821 0.3745 0.4129 g.3707 0.3669 0,4052 0.3632 03594 0.3557 0.3936 04325 0.3520 D.3483 3.1 0.9990 0.9991 0.9993 0.9991 09992 0.9992 09992 D.9992 0.9993 0.9993 0.9991 0,9995 -02 0.4207 04602 0.4168 0,4090 0,4013 0.3974 0.3897 0.3859 3.2 0.9993 0,9994 0.9994 09994 0.9994 09996 0.9995 0.9995 0,9995 0.4562 0.4522 0.4483 04443 04404 04364 0.4286 04247 3.3 .9995 0,9995 0.9996 09996 0.9996 D.9996 0.9996 0.9997 0480 0476 D468 0.4840 0,9997 D.9997 0.500 0.4960 Пдоо 0.4880 0.4721 D4641 3.4 0.9997 0,9997 0,9997 09997 0.9997 09997 0.9997 0.9998 Note that the probabilities given in this table represent the area to the LEFT of the z-score. The area to the RIGHT of a z-score 1- the area to the LEFT of the z-score