To solve the problem, we need to find the probability that the normal random variable [tex]\(X\)[/tex] with a mean ([tex]\(\mu\)[/tex]) of 15 and a standard deviation ([tex]\(\sigma\)[/tex]) of 10 will have a negative value, i.e., when [tex]\(X < 0\)[/tex].
Here's the step-by-step solution:
1. Understanding the Normal Distribution:
- A normal random variable [tex]\(X\)[/tex] follows the bell-shaped curve where [tex]\(\mu = 15\)[/tex] and [tex]\(\sigma = 10\)[/tex].
- We are asked to find [tex]\(P(X < 0)\)[/tex].
2. Standard Normal Variable (Z-Score):
- The Z-score helps us standardize the normal variable to the standard normal distribution (which has a mean of 0 and a standard deviation of 1).
The Z-score formula is:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
We need to find the Z-score when [tex]\(X = 0\)[/tex].
3. Calculate the Z-Score for [tex]\(X = 0\)[/tex]:
[tex]\[
Z = \frac{0 - 15}{10} = \frac{-15}{10} = -1.5
\][/tex]
4. Find the Probability Using the Cumulative Distribution Function (CDF):
- The CDF of the standard normal distribution will give us the probability that [tex]\(Z\)[/tex] is less than a particular value.
- We need to find [tex]\(P(Z < -1.5)\)[/tex].
From the Z-table or using statistical software, the probability corresponding to a Z-score of -1.5 is approximately 0.0668.
Thus, the probability that [tex]\(X\)[/tex] will have a negative value ([tex]\(X < 0\)[/tex]) is approximately [tex]\(0.0668\)[/tex].
So, the correct answer is:
c. 0.0668
