Problema Solution
on a standardized test with normal distribution the mean is 75 and the standard deviation is six if 1200 students took the test approximately how many students would be expected to be scored between 59 and 81
Answer provided by our tutors
Let μ = 75 be the mean and σ = 6, the standard deviation of the normal distribution.
Transform the original variable (of score) x into a standard normal distribution variable z by the formula z = (x-μ)/σ which then can be used to solve the problem.
To find the value of z corresponding to x = 59, substitute μ = 75, σ = 6 and x = 59 in the formula, z = (x-μ)/σ
z = (59 - 75)/6 = - 16/6 = - 8/3 = -2.67
To find the value of z corresponding to x = 81, substitute μ = 75, σ = 6 and x = 81 in the formula, z = (x-μ)/σ
z = (81 - 75)/6 = 6/6 = 1
The area of the curve between z = -2.67 and z = 0 is 0.4962 = 49.62%.
The area of the curve between z = 0 and z = 1 is 0.3413 = 34.13%.
Area of the curve between z = - 2.67 and z = 1 is (0.4962 + 0.3413 ) = 0.8375
Number of students expected to score between 59 and 81 is 1200 × 0.8375 = 1005.