Question

1)In a standard normal distribution, what proportion of the data set is within 1.75 standard deviations away from the mean? Write your answer in percent and with 2 decimal places. (Answer: 91.99 (with margin: 0.01) please show your work.)

2)A 100-point test was administered to a statistics class. Jack's score was in the 80^th percentile. Suppose the statistics class has 120 students and all of them took the test, how many students scored lower than or equal to Jack's score?(Answer:96.0 (with margin: 0.0 please show your work.)

3) 12 18 22 23 24 25 26 27 29 30 32 33

Determine the Upper fence.(Answer: 40.0 (with margin: 0.0) and 41.0 (with margin: 0.0) please show your work.)

Answer 1

Solution:

Question 1) :

X ~ Standard Normal distribution with mean =
μ=0
and standard deviation = $\sigma = 1$

We have to find proportion of the data set is within 1.75 standard deviations away from the mean.

That is: we have to find:

Pl-1.75 < X-μ < 1.75) = ?

1.75 X-μ 1.75 P(-1.75 < X-μ < 1.75)-P(

1.75 1.75 P(-1.75 < X-μ < 1.75)-P(

$P( -1.75 < X - \mu < 1.75) = P( -1.75 <Z< 1.75 )$

$P( -1.75 < X - \mu < 1.75) = P( Z< 1.75 ) - P( Z < -1.75 )$

Look in z table for z = 1.7 and 0.05 as well as for z= -1.7 and 0.05 and find area.

01 02 03 .04 05 06 .07 09 3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002 -3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003 3.2 .0007 0007 .0006 .0006 0006 .0006 .0006 .0005 .0005 .0005 -3.1 .0010 0009 0009 0 .0007 -3.0 .0013 0013 .0013 .0012 0012 .0011 .0011 .0011 .0010 .0010 0009 0008 0008 0008 .0008 .0007 0017 .0016 0016 .0015 .0015 0014 -2.8 .0026 0025 .0024 0023 0023 .0022 .0021 .0021 0020 0019 -2.7 .0035 .0034 .0033 0032 0031 0030 .0029 0028 .0027 0026 0036 -2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048 -2.4 .0082 0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 0064 0104 .0102 0099 0096 0094 0091 .0089 .0087 .0084 2.2 .0139 .0136 .0132 .0129 .012 .0122 .0119 .01 16 .0113 .0110 -2.1 .0179 0174 .0170 .0 .0143 0183 -1.9 .0287 .0281 .0274 0268 .0262 .0256 .0250 0244 0239 .0233 301 0294 1.7 .0446 .0436 .0427 0418 0409 0401 0392 .0384 .0375 .0367 0043 .0041 0040 .0047 .0045 .0044 0 .0039 0038 0037 0166 .0162 .0158 .0154 0150 .0146 -2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 1.8 0359 .0351 .0344 .0336 0329 0322 0314 0307 03

P( Z < -1.75) = 0.0401

200 .010203.04 .05 .06 .07 08.09 0.0 .5000 5040 5080 .5120 .5160 .519 0.1 5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 0.2 .5793 .5832 .5871 .5910 .5948 .598기 .6026 .6064 .6103 .6141 0.3 .6179 6217 .6255 6293 .6331 6368 6406 .6443 .6480 .6517 0.4 .6554 6591 .6628 .6664 6700 673 0.5 0.6 .7257 .7291 7324 735 7389 7472 7454 7486 7517 7549 0.7 .7580 0.8 7881 .7910 .7939 .7967 7995 .8 0.9 8159 .8186 .8212 8238 8264 1.0 8413 .8438 .8461 8485 .8508 1.1 8643 8665 .8686 8708 .87298749 8770 8790 8810 8830 1.2 .8849 .8869 8888 87 8925 944 8962 8980 .8997 9015 1.3 9032 .9049 .9066 .9082 9099 115 .9131 9147 .9162 .9177 1.4 .9192 .9207 9222 9236 9251 1.5 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441 1.6 9452 1.7 .5239 .5279 .5319 .5359 6772 .6808 .6844 .6879 7123 7157 7190 7224 6915 .6950 6985 .7019 .7054 .7088 7 7611 7642 7673 7704 7784 7764 7794 7823 .7852 3 .8051 .8078 .8106 .8133 289 . .8315 8340 8365 8389 531 .8554 8577.8599 .8621 8 265 .9279 9292 9306 9319 9463 9474 9484 9495 05 9515 9525 9535 9545 9599 9608 .9616 9625 9633

P( Z< 1.75) = 0.9599

Thus

$P( -1.75 < X - \mu < 1.75) = P( Z< 1.75 ) - P( Z < -1.75 )$

P(-1.75 < X-μ < 1.75) = 0.9599-0.0401

$P( -1.75 < X - \mu < 1.75) =0.9198$

To get exact answer as 0.9199, we need to use Calculator or Excel command:

Excel command is:

=NORM.S.DIST( 1.75 , TRUE) - NORM.S.DIST( - 1.75 , TRUE)

= 0.9199