Let x be a continuous random variable that is normally distributed with a mean of 65 and a standard deviation of 15. Find the probability that x assumes a value less than 45.
To find the probability that x assumes a value less than 45, we can use the standard normal distribution.
First, we need to standardize the random variable x to transform it into a standard normal distribution. We can do this using the formula:
Z = (X - μ) / σ
Where Z is the standardized value, X is the random variable, μ is the mean, and σ is the standard deviation.
In this case, we have:
X = 45,
μ = 65, and
σ = 15.
Substituting the values into the formula, we get:
Z = (45 - 65) / 15 = -20 / 15 = -4/3
Now, we can consult the standard normal distribution table or use a calculator to find the probability associated with the standardized value Z.
The standard normal distribution table provides the probabilities for values between 0 and Z. However, since we want to find the probability for a value less than 45, which corresponds to a Z-value less than -4/3, we need to find the area under the curve to the left of -4/3.
Using the table or a calculator, we find that the probability for a Z-value of -4/3 is approximately 0.0918.
So, the probability that x assumes a value less than 45 is approximately 0.0918 (or 9.18%).