X

 is a normally distributed random variable with mean 14 and standard deviation 16.

What is the probability that 

X

 is between 50 and 58?

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

Submit


To find the probability that a normally distributed random variable \( X \) is between 50 and 58, given that the mean \( \mu = 14 \) and the standard deviation \( \sigma = 16 \), we need to standardize these values to the standard normal distribution (with mean 0 and standard deviation 1).

### Step 1: Convert to Z-scores
The Z-score is calculated using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]

1. **For \( X = 50 \):**
   \[
   Z_1 = \frac{50 - 14}{16} = \frac{36}{16} = 2.25
   \]

2. **For \( X = 58 \):**
   \[
   Z_2 = \frac{58 - 14}{16} = \frac{44}{16} = 2.75
   \]

### Step 2: Find the probabilities for these Z-scores
Using standard normal distribution tables or a calculator:

- \( P(Z < 2.25) \approx 0.9878 \)
- \( P(Z < 2.75) \approx 0.9970 \)

### Step 3: Find the probability that \( Z \) is between 2.25 and 2.75
\[
P(2.25 < Z < 2.75) = P(Z < 2.75) - P(Z < 2.25) = 0.9970 - 0.9878 = 0.0092
\]

### Final Answer
The probability that \( X \) is between 50 and 58 is approximately **0.009** (rounded to the nearest thousandth).

Would you like more details or have any questions?

### Related Questions:
1. How do you find probabilities for different intervals in a normal distribution?
2. What is the significance of standardizing a normal random variable?
3. How do you calculate Z-scores for values outside the mean?
4. What are some applications of normal distribution in real life?
5. How would changing the standard deviation to 8 affect the probability between 50 and 58?

### Tip:
Remember, Z-scores allow you to standardize different normal distributions so you can use standard normal tables.

X is a normally distributed random variable with mean 14 and standard deviation 16. What is the probability that X is between 50 and 58? Write your answer as a decimal rounded to the nearest thousandth.

Find the probability that a normally distributed variable X with a mean of 14 and standard deviation of 16 is between 50 and 58. This problem involves key probability concepts like Z-scores and standard normal distribution, ideal for students in Grades 10-12.

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