Question

A makeup test is given and the average (μX) score out of 100 was 85.0, with a SD (σX) of 3.0. Assuming a normal distribution, find the dividing line (test scores) between the A's, B's, C's, D's, and E's. This time the highest 6% will be the A's, the next 16% B's, the next 26% C's, the next 36% D's.

Answer 1

Given Mean X = 85

Standard deviation X = 3

Assuming a normal distribution, find the dividing line (test scores) between the A's, B's, C's, D's, and E's. This time the highest 6% will be the A's, the next 16% B's, the next 26% C's, the next 36% D's.

A's from B's

We will find the test score between A's and B's

Since the highest 6% are be the A's, then we know that other 94% are less than A's. We can determine the z-score from the z-table using the p-value which is the area to the left of z-score and then find the respective test scores

Here p-value is 0.94 which is the area to the left of z-score

I have attached the z-tables at the end. We will calculate the z-scores using p-values from the respective table

In the positve z-table the area nearest to 0.94 is 0.93943 which is for a z-score of 1.55

The exact z-score for a p-value of 0.94 is 1.5548

We know that

Z-score = (X - X ) / X

1.5548 = (X - 85) / 3

X - 85 = 4.6644

X = 89.6644

X = 89.664 rounded to three dcimal places

So a test score of 89.664 differentiate the A's from B's

B's from C's

We will find the test score between B's and C's

Since the highest 6% are be the A's, the next 16% are B's, then we know that other 22% are less than B's. We can determine the z-score from the z-table using the p-value which is the area to the left of z-score and then find the respective test scores

Here p-value is 0.78 which is the area to the left of z-score

I have attached the z-tables at the end. We will calculate the z-scores using p-values from the respective table

In the positve z-table the area nearest to 0.78 is 0.7735 which is for a z-score of 0.77

The exact z-score for a p-value of 0.77219

We know that

Z-score = (X - X ) / X

0.7722 = (X - 85) / 3

X - 85 = 2.3166

X = 87.3166

X = 87.317 rounded to three dcimal places

So a test score of 87.317 differentiate the B's from C's

C's from D's

We will find the test score between C's and D's

Since the highest 6% are be the A's, the next 16% are B's, the next 26% are C's, then we know that other 52% are less than D's. We can determine the z-score from the z-table using the p-value which is the area to the left of z-score and then find the respective test scores

Here p-value is 0.52 which is the area to the left of z-score

I have attached the z-tables at the end. We will calculate the z-scores using p-values from the respective table

In the positve z-table the area nearest to 0.52 is 0.51994 which is for a z-score of 0.05

The exact z-score for a p-value of 0.05012

We know that

Z-score = (X - X ) / X

0.05012 = (X - 85) / 3

X - 85 = 0.1536

X = 85.15036

X = 85.1503 rounded to four dcimal places

So a test score of 85.1503 differentiate the C's from D's

D's from E's

We will find the test score between D's and E's

Since the highest 6% are be the A's, the next 16% are B's, the next 26% are C's, the next 36% are D's then we know that other 84% are less than E's We can determine the z-score from the z-table using the p-value which is the area to the left of z-score and then find the respective test scores

Here p-value is 0.16 which is the area to the left of z-score

I have attached the z-tables at the end. We will calculate the z-scores using p-values from the respective table

In the negative z-table the area nearest to 0.16 is 0.16109 which is for a z-score of 0.99

The exact z-score for a p-value of 0.99496

We know that

Z-score = (X - X ) / X

-0.99496 = (X - 85) / 3

X - 85 = -2.98488

X = 82.0151

X = 82.0151 7rounded to three dcimal places

So a test score of 82.015 differentiate the D's from E's

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-0.2 -0.3 0.4 -0.5 -0.6 0.7 -0.8 -0.9 -1.1 -1.2 -1.3 -1.4 -1.5 -1.6 .1.7 -1.8 -1.9 -2 -2.1 -2.2 -2.3 -2.4 -2.5 -2.6 -2.7 -2.8 -2.9 | 0.05 0.06 0.07 0.08 . 0.09 .50000 .49601 4920248803 .48405 480064760847210 .46812 .46414 .46017456204522444828 44433440344354043251 .42858 .42465 .42074 .41683 .41294 .40905 .40517 .40129 .3974339358 .38974 38591 .3820937828 .7448 .37070 .3669336317 .35942 .35569 .3519734827 344583409033724 .33350 .32997 .32636 .32276 .31918 .31561 .31207 30854305033013329806 29460 29116 28774 28434 2809627760 2742527093 26763 26435261092578525453251432482524510 .2419623885 23576.23270229652266322363220652177021476 | 211862089720611 2032720045 1966 19489 .1921518943 .18673 1840518141 1787917619 17361 17106 1685316602 .1635416109 .158561562515386.1515114917.146861445714231 .1400713786 .13567133501313612924 12714 1250712302.121001190011702 1150711314111231093510749.10565.10383102041002709853 . .09680 .09510 .0934209176 .09012 .08851 .08692 .0853408379 .08226 .08076 .0792707780 .0763607493 .0735307215 .07078 .0694406811 .06681 .06552 .06426 .0530106178 .060570593805821 .05705 .05592 .05480 .05370 .0526205155 .050500494704846 0474604648 .04551 .04457 .0436304272 .0418204093 04006 .03920 .03836 .03754 .03673 035903515034380336203288.03216031440307403005.02938 .0244202385 .02330 .02275022220215902118020680201801970 .01923 01876 01831 .0178601743 .01700 01659 01618 .01578 .01539 .01500 .01463 .01426 .01390 .0135501321012870125501222 0119101160 .0113001101 .01072 .01044 .01017 00990 .00964 00939 .60914 .0088900866 . .00842 .00820 .007980076.00755 .00734 .001400695 .00676 00557 .00639 .00621 .00604 .00587 057000554 .00539 .0052300508 .00494 . .00480 .00466 .00453 00440 .00427 .0041500402 .00391 .00379 .0036800357 .00347 .0033600326 .00317 .00307 .00298 .00289 .00280 .00272 .00254 .00256 .00248 .00240 .00233 .00226 .00219 .00212 .00205 0 0199 .00193 .00187 .0018100175 .00169 .0016400159 .00154 .00149 00149 .00144 .00139 .00135 .00131 .00126 .00122 .00118 .00114 00111 00107 .00104 .00100 .00097 .00094 .00090 .00087. .00084 .00082 .00079 .00076 00074. .00071 .00069 .0006600064 .00062 .00060 .000580005600054 .00052 .00050 .00048 ,00047 .0004500043 00042 ,000400003900038 .00036 .00035 .00034 .0003200031 .0000 .00029 .00028 .00027 .00026 .0002500024 .00023 .00022 00022 .0002100020 .00019 .00019 .00018 .00017 00017 .00016 .00015 .00015 .00014 .00014 .00013 .00013 0 0012 .00012 .00011 .000110001000010.000100000900009.000080000800008 .00008 .00007 .00007 .0000700006 .00006 .00006 .00006 .00005 .00005 .00005 .00005 .00005 .0000400004 .00004 .0000400004 .00004 .00003 .00003 .00003 .00003 .00003 .00003 .00003 .00003 .00002 .00002 .00002 .00002 -3.1 -3.2 -3.3 -3.4 -3.5 -3.6 -3.7 -3.8 -3.9