Question

Scores on an exam are normally distributed with a mean of 76 and a standard deviation of 10. In a group of 230 tests, how many students score below 66

Answer 1

On a bell curve (normal distribution), 68% of data points lie within 1 std deviation, 95% within 2 std deviations, and 99% within 3 std deviations. Since a score of 66 is 1 std deviation away to the left, then to calculate the number of students below 66:

64% accounts for 1 std deviation to the left and right, but since we are going left, then 64%/2 = 34% of students are 1 std deviation away to the left from the mean. Given that the mean is 76 and the mean is the center of the bell curve, then 50% are lower than 76 and 50% are higher than 76. With this information we can calculate the percentage of students that are below 1 std deviation to the left:

50% - 34% = 16%. So 16% or 0.16 percent of students got a grade lower than 66. Given that you have 230 students: 230 * .16 = 37

Answer 2

you can use a calculator
normalcdf ( -1E99, 66, 76, 10)

Answer 3

that gives you the percentage, then multiply by 230 , that gives you 36.49 students