Assume the random variable x is normally distributed with mean muequals82 and standard deviation sigmaequals4. Find the indicated probability.
​P(xless than74​)
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Part 1
​P(xless than74​)equals
  
enter your response here ​(Round to four decimal places as​ needed.)

To solve for the probability \( P(x < 74) \) given that \( x \) is normally distributed with a mean \( \mu = 82 \) and a standard deviation \( \sigma = 4 \), we can follow these steps:

1. **Standardize the variable**: Convert the value \( x = 74 \) to its corresponding \( z \)-score using the formula:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   Substituting the given values:
   \[
   z = \frac{74 - 82}{4} = \frac{-8}{4} = -2
   \]

2. **Find the probability**: Now, we need to find the probability that \( z \) is less than \(-2\). This is denoted by \( P(z < -2) \).

3. **Use the standard normal distribution table**: Look up the value of \( P(z < -2) \) in the standard normal distribution table, or use a calculator that provides this functionality.

   From the standard normal distribution table:
   \[
   P(z < -2) \approx 0.0228
   \]

So, the probability \( P(x < 74) \) is approximately **0.0228**.

Would you like more details on this calculation or have any questions?

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Here are 8 related questions you might find useful:

1. What is the probability \( P(x > 90) \) for the same distribution?
2. How would you find \( P(74 < x < 90) \) using the normal distribution?
3. Can you calculate the probability \( P(x < 82) \) for this distribution?
4. What does the \( z \)-score represent in a normal distribution?
5. How would the probability change if the standard deviation were \( \sigma = 6 \)?
6. What is the cumulative distribution function (CDF) for a normal distribution?
7. How do you find the inverse of a probability (i.e., find \( x \) for a given probability)?
8. How does changing the mean \( \mu \) affect the shape of the normal distribution?

**Tip**: When dealing with normally distributed variables, understanding how to standardize to a \( z \)-score is key to solving many probability problems.

Learn how to calculate the probability ( P(x < 74) ) for a normal distribution with mean μ = 82 and standard deviation σ = 4. Follow a step-by-step solution including standardizing to z-score and using the standard normal distribution table. Suitable for advanced high school students and those interested in probability and statistics.

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