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To solve this problem, we need to calculate the probability for a given range in a normal distribution, specifically \( P(10 \leq x \leq 26) \), where \( x \) follows a normal distribution with mean \( \mu = 15.8 \) and standard deviation \( \sigma = 4.3 \).

Here's the step-by-step solution:

1. **Convert to Z-scores**:
   The Z-score formula is:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   
   Calculate the Z-scores for \( X = 10 \) and \( X = 26 \):
   
   - For \( X = 10 \):
     \[
     Z_1 = \frac{10 - 15.8}{4.3} = \frac{-5.8}{4.3} \approx -1.3488
     \]
   
   - For \( X = 26 \):
     \[
     Z_2 = \frac{26 - 15.8}{4.3} = \frac{10.2}{4.3} \approx 2.3721
     \]

2. **Find Probabilities**:
   Using standard normal distribution tables or a calculator:
   
   - \( P(Z \leq -1.3488) \approx 0.0884 \)
   - \( P(Z \leq 2.3721) \approx 0.9911 \)

3. **Calculate the Probability**:
   Now, the probability for \( 10 \leq X \leq 26 \) is:
   \[
   P(10 \leq X \leq 26) = P(Z \leq 2.3721) - P(Z \leq -1.3488)
   \]
   Substituting the values:
   \[
   P(10 \leq X \leq 26) = 0.9911 - 0.0884 = 0.9027
   \]

### Answer:
\[
P(10 \leq x \leq 26) \approx 0.9027
\]

Would you like more details on any part of this solution?

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Here are five related questions to deepen understanding:

1. How do you interpret a Z-score in a normal distribution?
2. How would you calculate \( P(x \geq 26) \) with the same mean and standard deviation?
3. What is the significance of a probability close to 1 in a normal distribution?
4. How would the answer change if the mean were higher, say \( \mu = 20 \)?
5. Can you explain how the shape of the normal distribution affects probabilities?

**Tip:** For normal distribution problems, remember to convert values to Z-scores to easily find probabilities using the standard normal table.

Assume that x has a normal distribution with a mean of μ = 15.8 and a standard deviation of σ = 4.3. Find the probability P(10 ≤ x ≤ 26).

Learn how to calculate the probability P(10 ≤ x ≤ 26) for a normal distribution with a mean of 15.8 and a standard deviation of 4.3. This detailed guide covers Z-score calculations, standard normal distribution, and step-by-step probability determination. Suitable for high school students in Grades 11-12.

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