To find the indicated probability for a normally distributed random variable $x$ with a given mean $\mu$ and standard deviation $\sigma$, you can follow these steps:

1. Standardize the random variable: Convert the given $x$ value to a standard normal variable $z$ using the formula: $z = \frac{x - \mu}{\sigma}$

2. Use the z-table: Find the corresponding probability from the z-table for the computed $z$-score.

Let's assume we need to find the probability $P(X \leq x)$ for a specific value of $x$. For demonstration purposes, let’s find the probability for $x = 90$.

Given:

• Mean ($\mu$) = 88
• Standard deviation ($\sigma$) = 4

Step-by-Step Solution:

1. Standardize the $x$ value: $z = \frac{x - \mu}{\sigma} = \frac{90 - 88}{4} = \frac{2}{4} = 0.5$

2. Find the corresponding probability:

   • Look up the $z$-score of 0.5 in the z-table.
   • The cumulative probability for $z = 0.5$ is approximately 0.6915.

Therefore, the probability $P(X \leq 90)$ is approximately 0.6915.

If you have a specific $x$ value in mind, please provide that value, and I can find the corresponding probability for you.

Do you want more details or have any questions? Here are some related questions for further exploration:

1. What is the probability $P(X \leq 85)$ for the given normal distribution?
2. How do you find the probability $P(X > 92)$?
3. What is the probability $P(86 \leq X \leq 90)$?
4. How do you use the standard normal distribution to find probabilities for non-standard normal distributions?
5. What is the probability that $X$ is within one standard deviation of the mean?
6. Can you explain the empirical rule (68-95-99.7 rule) for normal distributions?
7. How do you find the $x$ value corresponding to a given percentile in a normal distribution?
8. What is the probability $P(X < 80)$ for the given normal distribution?

Tip: When working with probabilities for normal distributions, always convert your raw scores to $z$-scores to use the standard normal table effectively.