Given: Mean(μ) = 2, Standard Deviation(σ) = 2, 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 0.50:
P (2 < X < 3) = P (0.00 < Z < 0.50) = P (Z < 0.50) – P (Z < 0.00)
From a standard normal distribution table (values rounded to five decimal places):
P (Z < 0.50) = 0.69146
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 2 is between 2 and 3:
P (2 < X < 3) = P (0.00 < Z < 0.50) = 0.69146 – 0.5 = 0.19146
P (2 < X < 3) = 0.19146 = 19.146%
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 – 2/2
= 0/2
= 0.00Z3 =
X2 – μ/σ
= 3 – 2/2
= 1/2
= 0.50Therefore, the Probability that X is between 2 and 3 equals the Probability that Z is between 0.00 and 0.50:
P (2 < X < 3) = P (0.00 < Z < 0.50) = P (Z < 0.50) – P (Z < 0.00)
From a standard normal distribution table (values rounded to five decimal places):
P (Z < 0.50) = 0.69146
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 2 is between 2 and 3:
P (2 < X < 3) = P (0.00 < Z < 0.50) = 0.69146 – 0.5 = 0.19146
P (2 < X < 3) = 0.19146 = 19.146%
