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

Since X is a normal random variable, we know that the area under the curve to the left of 5.54 is equal to the cumulative probability up to that point. From the diagram, we see that this probability is 0.67.

Using a table or calculator, we can find that the z-score corresponding to a cumulative probability of 0.67 is approximately 0.44.

We can use the formula for a standardized normal variable to find the value of the random variable X:

z = (X - mean) / standard deviation

0.44 = (5.54 - 4) / standard deviation

Simplifying and solving for the standard deviation:

standard deviation = (5.54 - 4) / 0.44 = 3.5

Therefore, the standard deviation of X is 3.5, which is answer choice B.

If X Is a Normal Random Variable with Mean 4