Given: Mean(μ) = 2, Standard Deviation(σ) = 1, Data Point(X1) = 2, Data Point(X2) = 3
If X is a normally distributed random variable X~N(μ, σ2), then
Since Z , the standard normal distribution is also N(0, 1), we standardize X by finding the Z-scores corresponding to X1 = 2 and X2 = 3:
Z2 =
Z3 =
Therefore, the Probability that X is between 2 and 3 equals the Probability that Z is between 0.00 and 1.00:
P (2 < X < 3) = P (0.00 < Z < 1.00) = P (Z < 1.00) – P (Z < 0.00)
From a standard normal distribution table (values rounded to five decimal places):
P (Z < 1.00) = 0.84134
P (Z < 0.00) = 0.5
As a result, the Probability that X, a normally distributed random variable with a mean of 2 and a standard deviation of 1 is between 2 and 3:
P (2 < X < 3) = P (0.00 < Z < 1.00) = 0.84134 – 0.5 = 0.34134
P (2 < X < 3) = 0.34134 = 34.134%
Note: P (2 < X < 3) is equal to the Probability that X is between the Mean and 1 Standard Deviation above the Mean
If X is a normally distributed random variable X~N(μ, σ2), then
X – μ/σ
is N(0, 1)Since Z , the standard normal distribution is also N(0, 1), we standardize X by finding the Z-scores corresponding to X1 = 2 and X2 = 3:
Z2 =
X1 – μ/σ
= 2 – 2/1
= 0/1
= 0.00Z3 =
X2 – μ/σ
= 3 – 2/1
= 1/1
= 1.00Therefore, the Probability that X is between 2 and 3 equals the Probability that Z is between 0.00 and 1.00:
P (2 < X < 3) = P (0.00 < Z < 1.00) = P (Z < 1.00) – P (Z < 0.00)
From a standard normal distribution table (values rounded to five decimal places):
P (Z < 1.00) = 0.84134
P (Z < 0.00) = 0.5
As a result, the Probability that X, a normally distributed random variable with a mean of 2 and a standard deviation of 1 is between 2 and 3:
P (2 < X < 3) = P (0.00 < Z < 1.00) = 0.84134 – 0.5 = 0.34134
P (2 < X < 3) = 0.34134 = 34.134%
Note: P (2 < X < 3) is equal to the Probability that X is between the Mean and 1 Standard Deviation above the Mean
