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X is a normally distributed random variable with mean 25 and standard deviation 8.
What is the probability that X is between 1 and 17?
Use the 0.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

To solve this problem, we can use the 68-95-99.7 rule, also known as the empirical rule, which applies to normal distributions.

### Step 1: Understand the Problem
- **Mean (μ)**: 25
- **Standard Deviation (σ)**: 8
- We want to find the probability that \( X \) is between 1 and 17.

### Step 2: Calculate the Z-scores
The Z-score for a value \( X \) in a normal distribution is given by:
\[
Z = \frac{X - \mu}{\sigma}
\]
- For \( X = 1 \):
\[
Z_1 = \frac{1 - 25}{8} = \frac{-24}{8} = -3
\]
- For \( X = 17 \):
\[
Z_2 = \frac{17 - 25}{8} = \frac{-8}{8} = -1
\]

### Step 3: Apply the Empirical Rule
- **Z = -1**: This corresponds to 1 standard deviation below the mean. The empirical rule tells us that about 68% of the data lies within 1 standard deviation of the mean, so from \( Z = -1 \) to \( Z = 1 \), the probability is 68%. The area to the left of \( Z = -1 \) is therefore \( \frac{1 - 0.68}{2} = 0.16 \) or 16%.
  
- **Z = -3**: This corresponds to 3 standard deviations below the mean. According to the empirical rule, 99.7% of the data lies within 3 standard deviations of the mean. Therefore, the area to the left of \( Z = -3 \) is \( \frac{1 - 0.997}{2} = 0.0015 \) or 0.15%.

### Step 4: Find the Probability Between Z = -3 and Z = -1
To find the probability that \( X \) is between 1 and 17, subtract the area to the left of \( Z = -3 \) from the area to the left of \( Z = -1 \):
\[
P(-3 < Z < -1) = P(Z < -1) - P(Z < -3) = 0.16 - 0.0015 = 0.1585
\]

### Final Answer
The probability that \( X \) is between 1 and 17 is approximately **0.159** (rounded to the nearest thousandth).

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the probability that \( X \) is between 25 and 41?
2. What is the probability that \( X \) is greater than 33?
3. If the mean changes to 30 but the standard deviation remains the same, what is the probability that \( X \) is between 10 and 22?
4. Calculate the Z-score for \( X = 9 \) with the same mean and standard deviation.
5. What is the probability that \( X \) is less than 9?

**Tip:** The empirical rule is a quick way to estimate probabilities in a normal distribution, but for more precise calculations, use Z-tables or statistical software.

Learn how to calculate the probability that a normally distributed random variable X with mean 25 and standard deviation 8 falls between 1 and 17. Understand the Z-score calculation and application of the empirical rule (68-95-99.7 rule) in statistics.

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