In normal distribution, 31% of the items are under 35 and 12% are over 62. Find the mean
and standard deviation of the distribution.

To find the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the normal distribution, we can use the fact that in a normal distribution, the percentages provided correspond to certain z-scores.

### Given:
- 31% of the items are under 35.
- 12% of the items are over 62.

### Steps:

1. **Identify the z-scores associated with the given percentages:**
   - For \(31\%\) of the items under 35, this implies \(P(X \leq 35) = 0.31\). From the z-table (or using a calculator), the corresponding z-score is approximately \(-0.50\).
   - For \(12\%\) of the items over 62, this implies \(P(X \geq 62) = 0.12\). This means \(P(X \leq 62) = 1 - 0.12 = 0.88\). From the z-table, the corresponding z-score is approximately \(1.18\).

2. **Set up the z-score formulas:**
   The z-score formula is:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

   Using this formula for both given points:

   - For \(X = 35\), \(z = -0.50\):
     \[
     -0.50 = \frac{35 - \mu}{\sigma}
     \]
   - For \(X = 62\), \(z = 1.18\):
     \[
     1.18 = \frac{62 - \mu}{\sigma}
     \]

3. **Solve the system of equations:**
   - From the first equation:
     \[
     -0.50 = \frac{35 - \mu}{\sigma} \implies 35 - \mu = -0.50\sigma
     \]
     \[
     \mu = 35 + 0.50\sigma \tag{1}
     \]
   - From the second equation:
     \[
     1.18 = \frac{62 - \mu}{\sigma} \implies 62 - \mu = 1.18\sigma
     \]
     \[
     \mu = 62 - 1.18\sigma \tag{2}
     \]

4. **Substitute and solve for \(\sigma\):**
   Equating equations (1) and (2):
   \[
   35 + 0.50\sigma = 62 - 1.18\sigma
   \]
   Combine like terms:
   \[
   1.68\sigma = 27
   \]
   Solve for \(\sigma\):
   \[
   \sigma = \frac{27}{1.68} \approx 16.07
   \]

5. **Find \(\mu\):**
   Substitute \(\sigma = 16.07\) into equation (1):
   \[
   \mu = 35 + 0.50(16.07) \approx 35 + 8.04 = 43.04
   \]

### Final Answer:
- The **mean** \(\mu \approx 43.04\)
- The **standard deviation** \(\sigma \approx 16.07\)

Let me know if you'd like more details or have any questions.

Here are 5 related questions to expand your understanding:
1. How would the values change if 50% of the items were under 35?
2. What is the probability that a randomly selected item falls between 35 and 62?
3. If the standard deviation was smaller, say 10, how would that impact the distribution?
4. How do z-scores help in comparing different normal distributions?
5. What would be the z-score if the value is 50?

**Tip**: Z-scores are a great tool for understanding how far a value is from the mean in terms of standard deviations!

In normal distribution, 31% of the items are under 35 and 12% are over 62. Find the mean and standard deviation of the distribution.

Learn how to find the mean and standard deviation of a normal distribution, given that 31% of the items are under 35 and 12% are over 62. This step-by-step guide explores key statistical concepts like z-scores and the properties of normal distribution. Suitable for students in Grades 10-12.

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