Question

A mandatory competency test for high school sophomores has a normal distribution with a mean of 400 and a standard deviation of 100.

a) The top 3% of students receive $500. What is the minimum score you would need to receive this award?

b) The bottom 1.5% of students must go to summer school. What is the minimum score you would need to stay out of this group?

Answer 1

solution:

the distribution given as normal with the following mean and standard deviation

u= 400,0 = 100

a)

proportion of top students who receive $500 = 3% = 0.03

so area to the right of those who receive reward = 0.03

now we have to find the corresponding z score with (1-0.03) = 0.97 area to the left

the z score from the z table with 0.97 area to the left = 1.88

so $z=\frac{X-\mu}{\sigma}$

$X = (z*\sigma)+\mu=(1.88*100)+400=588$

so the minimum score to get the award is 588

b)

proportion of bottom 1.5% need to go to summer school

z score which corresponds the area of 0.015 to the left is -2.17

so,
X = (2+0) +u=(-2.17 * 100) + 400 = 183

so minimum score require to out of this group is 183

deminstration of z score finding from z table as follows

table represents PIZSZ) .0001 oo13 .0012 0010 oid 1-1# ཤཱ ཉྙཱཀྐ ཀྐཌྷ ཌྷ གྷ བྷཱབྷཱཐཱ ཡྻོ ཙྩ བྷ ཝཱབྷཱཐཱ ཀྐ བྷ ， 0.000.01 0. 2030040.050.060. 0. 00 .09 .0002 .0002 .0001 .0001 .000 .0001 .0001 .0001 .0001 .0002 .0002 .0002 0 002 .0002 .0002 .0002 -0002 .0002 .0002 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 ,0002 .0005 ,0005 .0005 .0004 .0004 .0004 .00040004 .0004 .0003 .0007 ,0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005 .0010 .0009 .0009 .0009 .0008 .008 .0008 .000B .0007 .0007 .0013 .0013 .0012 .0011 .0011 .0011 .0019 ,001 .001 .0017 .00i6 ། .0016 .0015 .0015 .0014 .0014 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0026 .0047 .0044 ,0043 .0041 .0040 .0039 .0038 .0037 .0062 0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048 .0082 .0080 .0078 .0075 .0073 .007i .0069 .0068 .0066 .0064 -23.0107 0i04 01020099 .0096 .0094 .0091 .0089 .0087 .0084 1201390136 .0132 .0129 .0125 .01220119 .0116 40113 .00 .0179 .0174 ,0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143 -20 |.0228 .0222027 ,0212 .02070202 .017 .0192018 .0183 =102802a102740268 .026202560250 .0244 .0239 .0359035། 0344036.03230322081403070301.0294 0027 ,0045 .0036

PIZS2) ai 1 17157 0 .00 0.00 0.0 0. 8 0 .00 0.6 1.5000 504050805120 - 5160 5199 -5398 5438 5478 5517 5557 .5596 5793 | 5832 1.5871 5910 5948 5987 6179 6217 .6255 .6293 6331 .6366 6554 1.6591 1.6628 6664 .6700 6736 46915 1,6950 6985 1.7019 7054 .70BB 7257 17291 17324 7357 7389 1.7422 17580 1.7611 17642 1.7673 7704 7734 1.7881 | 7910 4.793 .7967 1.7995 8023 18159 8186 .6212 82388264 .8269 8413 | 18438 8461 ,8485 1.8508 ,8531 8643 1,8665 ,8686 .8708 18729 .8749 18849 1.8869 1,888 1.8907 8925 1.8944 19032 1,9049 .9066 | 19082 9099 9115 19192 19207_ 9222 19236 9251 9265 9332 1.9345 9357 19370 19362 9394 9452 1.9463 9474 19484 19495 | 19505 19554 19564 9573 9582 9591 1.9599 19641 | 19649 9656 19664 19671 9678 9713 | 9719 19726 19732 19738 19744 9772 .9778 9783 9788 978979 19821 19825 19830 9889842 19861 9864 986 98719675 978 993 9896 9898 99019904 9906 0.06 0.07 15239 15279 15636 5675 16026 6064 6406 .6443 1672 .6808 17123 7454 1.7466 17764 47794 .8051 | 18078 8315_8340 18554 ,857 1.8770 .8790 18962 18980 19131 19147 19279 19292 19406 19418 19515 9525 9608 616 19666 1.9693 19750 19756 90 19808 | 9846 19850 19819 84 90 911 0. 5319 5714 6103 16460 16844 17190 1.7517 47823 1.8106 .8365 1.8599 1.8 10 18997 9162 19306 | 9429 19535 9625 19699 9761 19812 9854 987 913 0.0 5359 5753 6141 6517 6879 17224 17549 17852 ,8133 ,8389 18621 18830 19015 19177 19319 19441 | 19545 19633 19706 19767 | 9834 9817 9857 19890 916