The test Score of a Math class with 800 students are distributed
normally with a mean of 75 and a standard deviation of 7.
a) What percentage of a class has a test score between 68 and 82?
b) Approximately how many students have a test score between 61
and 89?
c) What is the probability that a student chosen at random has a
test score between 54 and 75?
d) Approximately how many students have a test score greater
than or equal to 96?

You want to know various percentages and numbers of students in different test score ranges when 800 students have scores normally distributed with a mean of 75 and a standard deviation of 7.

a) [68, 82]

The range 68 to 82 is one standard deviation either side of the mean: 75±7. The "empirical rule" tells you about 68% of students will have a test score in this range. An appropriate calculator gives the fraction as 68.3%.

b) [61, 89]

The range 61 to 89 is 2 standard deviations either side of the mean: 75±2·7. The "empirical rule" tells you about 95% of the 800 students will have a test score in this range. An appropriate calculator puts the number slightly higher than 760 students: 764 students.

c) [54, 75]

The range 54 to 75 is 3 standard deviations below the mean. The empirical rule tells you about 99.7% of the group falls inside 3 standard deviations. Half this number is below the mean, about 49.85% An appropriate calculator puts the number slightly higher: 49.87%.

d) [96, ∞)

The scores at 96 and above are more than 3 standard deviations above the mean. As in part (c), the number in that tail are given by the empirical rule as about 0.15%, or 1.2 students. An appropriate calculator puts the number in the tail of the distribution at a slightly lower value. Hence, the estimated number of students with a score at least 96 is 1 student.

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Additional comment

The numbers in this question seem to be chosen so you can use the empirical rule to find the answers. That rule will make the answers slightly different: 68%, 760 students, 49.85%, 1 student.