In an Algebra II class with 135 students, the final exam scores have a mean of 72.3 and standard deviation 6.5. The exams on the exams are whole numbers, and the grade pattern follows a normal curve.

a. Find the number of students who receive grades from two to three standard deviations above the mean? Explain how you got your answer!

b. Find the number of students who scored between 65.8 and 72.3. Explain how you found your answer!

Population = 135 students

Mean score = 72.3

Standard deviation of the scores = 6.5

Part (a) : Students from 2SD and 3SD above the mean

2SD below and above the mean includes 95% of the population while 3SD includes 99.7% of the population.

95% of population = 0.95*135 ≈ 129 students

99.7% of population = 0.997*135 ≈ 135 students

Therefore, number of students from 2SD to 3SD above and below the bean = 135 - 129 = 6 students.

In this regard, Students between 2SD and 3SD above the mean = 6/2 = 3 students

Part (b) : Students who scored between 65.8 and 72.3

The first step is to calculate Z values

That is,

Z = (mean-X) / SD

Z at 65.8 = (72.3-65.8) / 6.5 = 1

Z at 72.3 = (72.3-72.3) / 6.5 = 0

Second step is to find the percentages at the Z values from Z table.

That is,

Percentage of population at Z (65.8) = 0.8413 = 84.13%

Percentage of population at (Z (72.3) = 0.5 = 50%

Third step is to calculate number of students at each percentage.

That is,

At 84.13%, number of students = 0.8413*135 ≈ 114

At 50%, number of students = 0.5*135 ≈ 68

Therefore, students who scored between 65.8 and 72.3 = 114-68 = 46 students