Assume the random variable x is normally distributed with mean 

muμequals=8282 

and standard deviation 

sigmaσequals=44. 

Find the indicated probability.

​P(xless than<7474​)

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Part 1

​P(xless than<7474​)equals=enter your response here 

​(Round to four decimal places as​ needed.)

To find the probability \( P(X < 74) \) for a normally distributed random variable \( X \) with mean \( \mu = 82 \) and standard deviation \( \sigma = 4 \), we can use the standard normal distribution.

First, we need to convert the value \( x = 74 \) to a z-score using the formula:

\[ z = \frac{x - \mu}{\sigma} \]

Substituting the given values:

\[ z = \frac{74 - 82}{4} = \frac{-8}{4} = -2 \]

Now, we need to find the probability \( P(Z < -2) \) where \( Z \) is a standard normal variable (mean 0 and standard deviation 1).

Using standard normal distribution tables or a calculator, we find:

\[ P(Z < -2) \]

Looking up the z-score of -2.00 in the standard normal distribution table, we find that the corresponding probability is 0.0228.

Therefore, the probability \( P(X < 74) \) is:

\[ P(X < 74) = 0.0228 \]

So, the answer is:

\[ \boxed{0.0228} \]

Would you like more details or have any questions?

Here are some related questions to consider:

1. What is the probability that \( X \) is greater than 90?
2. Find the probability \( P(80 < X < 85) \).
3. Determine the z-score for \( X = 70 \).
4. Calculate \( P(X > 78) \).
5. What is the 95th percentile of this distribution?
6. Find the probability \( P(X = 82) \) (Hint: For continuous distributions, the probability at an exact point is 0).
7. If we take a sample of size 30, what is the standard error of the mean?
8. What is the probability that the sample mean of 30 observations is less than 80?

**Tip:** Always sketch the normal curve and shade the area of interest to visualize the problem better.

Learn how to calculate the probability that a normally distributed variable with mean 82 and standard deviation 4 is less than 74. Use the z-score formula and standard normal distribution tables for accurate results. Suitable for advanced high school or college students.

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