X

 is a normally distributed random variable with mean 80 and standard deviation 11.

What is the probability that 

X

 is between 3 and 61?

*Write* *your answer as a decimal rounded to the nearest* *thousandth.*

To find the probability that a normally distributed random variable \( X \) is between 3 and 61, given that the mean is 80 and the standard deviation is 11, we need to calculate the standard normal \( Z \)-scores for both 3 and 61.

The formula to calculate the \( Z \)-score is:

\[
Z = \frac{X - \mu}{\sigma}
\]

where:
- \( X \) is the value of the random variable,
- \( \mu \) is the mean, and
- \( \sigma \) is the standard deviation.

### Step 1: Calculate the \( Z \)-scores

For \( X = 3 \):
\[
Z_1 = \frac{3 - 80}{11} = \frac{-77}{11} = -7
\]

For \( X = 61 \):
\[
Z_2 = \frac{61 - 80}{11} = \frac{-19}{11} \approx -1.727
\]

### Step 2: Find the probabilities from the standard normal distribution

Now, using the standard normal distribution table or a calculator, we find the probabilities associated with these \( Z \)-scores.

For \( Z_1 = -7 \), the probability is nearly 0, since the value is extremely far in the negative tail of the distribution.

For \( Z_2 = -1.727 \), the cumulative probability is approximately 0.042.

### Step 3: Calculate the final probability

The probability that \( X \) is between 3 and 61 is:

\[
P(3 < X < 61) = P(Z_2) - P(Z_1) = 0.042 - 0 \approx 0.042
\]

Thus, the probability that \( X \) is between 3 and 61 is approximately **0.042**, rounded to the nearest thousandth.

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

Here are 5 related questions to consider:
1. What is the probability that \( X \) is less than 50 in this distribution?
2. How does changing the standard deviation affect the probability calculation?
3. What is the probability that \( X \) is greater than 100?
4. How would the probability change if the mean were 90 instead of 80?
5. How do we interpret the standard normal distribution in real-world applications?

**Tip:** When working with normal distributions, always calculate the \( Z \)-score to transform data into the standard normal distribution before finding probabilities.

X is a normally distributed random variable with mean 80 and standard deviation 11. What is the probability that X is between 3 and 61? Write your answer as a decimal rounded to the nearest thousandth.

Find the probability that a normally distributed variable X with a mean of 80 and standard deviation of 11 falls between 3 and 61. This problem demonstrates how to calculate Z-scores and use the standard normal distribution table to find the probability. Suitable for students in Grades 10-12.

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