\mu = MEAN and \sigma = STDDEV.
What fraction of test-takers had a grade between GRADES[0]
and GRADES[1] on the exam?
Use the cumulative z-table provided below.
A cumulative z-table shows the probability that a standard normal variable will be less than a certain value (z).
In order to use the z-table, we first need to determine the z-scores of the two grades which we were given.
Recall that we can calculate the z-scores by subtracting the
mean (\mu) from
each exam grade and then dividing by the standard
deviation (\sigma).
\large{\quad z \quad = \quad
\dfrac{x - \pink{\mu}}{\purple{\sigma}}
\quad = \quad \dfrac{grade - \pink{MEAN}}{\purple{STDDEV}}
\quad = \quad localeToFixed(ZSCORES[index], 2)}
We can find the percentage of test-takers that earned between
GRADES[0] and GRADES[1]
by finding the area between localeToFixed(ZSCORES[0], 2) and
localeToFixed(ZSCORES[1], 2) under the standard normal curve. After
looking up these two z-scores in our z-table, subtracting the two table
values will provide us with the total area.
Since the normal curve is symmetrical, the area less than
localeToFixed(ZSCORES[1], 2) is equal to the area
greater than localeToFixed(abs( ZSCORES[1] ), 2), which can
be found by looking up localeToFixed(abs( ZSCORES[1] ), 2)
on the z-table and subtracting the table value from 1, the
total area under the curve.
1 - localeToFixed(ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ], 4) =
localeToFixed(1 - ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ], 4).
This value,
localeToFixed(1 - ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ], 4),
represents the portion of the population that scored lower than
GRADES[1] on the exam.
Look up localeToFixed(ZSCORES[1], 2) on the z-table. This value,
localeToFixed(ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ], 4),
represents the portion of the population that scored lower than
GRADES[1] on the exam.
Since the normal curve is symmetrical, the area less than
localeToFixed(ZSCORES[0], 2) is equal to the area
greater than localeToFixed(abs(ZSCORES[0]), 2), which can
be found by looking up localeToFixed(abs(ZSCORES[0]), 2)
on the z-table and subtracting the table value from 1, the
total area under the curve.
1 - localeToFixed(ZGRID[ROW_INDEX_1][COL_INDEX_1], 4) =
localeToFixed(1 - ZGRID[ROW_INDEX_1][COL_INDEX_1], 4). This value,
\green{localeToFixed(1 - ZGRID[ROW_INDEX_1][COL_INDEX_1], 4)},
represents the portion of the population that scored lower than
GRADES[0] on the exam.
Look up localeToFixed(ZSCORES[0], 2) on the z-table. This value,
\green{localeToFixed(ZGRID[ ROW_INDEX_1 ][ COL_INDEX_1 ], 4)},
represents the portion of the population that scored lower than
GRADES[0] on the exam.
Subtract the two cumulative areas:\blue{
localeToFixed(1 - ZGRID[ROW_INDEX_2][COL_INDEX_2], 4)
localeToFixed(ZGRID[ROW_INDEX_2][COL_INDEX_2], 4)
} - \green{
localeToFixed(1 - ZGRID[ROW_INDEX_1][COL_INDEX_1], 4)
localeToFixed(ZGRID[ROW_INDEX_1][COL_INDEX_1], 4)
} = \pink{localeToFixed(ANSWER, 4)}
Thus, localeToFixed(ANSWER * 100, 2)\%
of the test-takers scored between GRADES[0] and
GRADES[1] on the course( 1 ) exam.