If X is a normal random variable with mean 4, and if the probability that X is less than 5.54 is .67 (as shown below), then what is the standard deviation of X? (Note: the diagram is not necessarily to scale.)

A)1.75
B)3.50
C)4.20
D)12.25

Question

If X is a normal random variable with mean 4, and if the probability that X is less than 5.54 is .67 (as shown below) , then what is the standard deviation of X? (Note: the diagram is not necessarily to scale.)

A)1.75 
B)3.50 
C)4.20 
D)12.25

Quizplus · Accepted Answer

From the normal distribution table, we can find the z-score associated with a probability of .67:
z = 0.44
Now we use the formula for a z-score:
z = (x - μ) / σ
0.44 = (5.54 - 4) / σ
Solving for σ, we get:
σ = (5.54 - 4) / 0.44 = 3.50
Therefore, the standard deviation of X is 3.50, which is answer choice B. We know that:

P(X < 5.54) = 0.67

We can standardize this using the z-score formula:

z = (5.54 - 4) / σ

where σ is the standard deviation of X.

Solving for σ:

0.67 = P(X < 5.54) = P(z < (5.54 - 4) / σ)

Using a standard normal distribution table or calculator, we can find that z = 0.44.

0.44 = (5.54 - 4) / σ

Solving for σ:

σ = (5.54 - 4) / 0.44 = 3.50

Therefore, the standard deviation of X is 3.50. Answer choice B is the correct choice.

If X Is a Normal Random Variable with Mean 4