In a Normal Distribution 7% of the items are under 35 and 89% are under 63 determine the mean and standard deviation of the distribution 

We are given two key pieces of information from the normal distribution:

1. 7% of the items are below 35.
2. 89% of the items are below 63.

The formula for converting a value \( x \) to a z-score in a normal distribution is:

\[
z = \frac{x - \mu}{\sigma}
\]

where:
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation,
- \( x \) is the value from the distribution, and
- \( z \) is the z-score corresponding to that value.

### Step 1: Find the z-scores corresponding to the given probabilities.
Using a standard normal distribution table (z-table) or a calculator:

- The z-score corresponding to 7% (or 0.07) is approximately \( z_1 = -1.475 \).
- The z-score corresponding to 89% (or 0.89) is approximately \( z_2 = 1.227 \).

### Step 2: Set up the system of equations.
We now have two equations based on the formula \( z = \frac{x - \mu}{\sigma} \):

\[
z_1 = \frac{35 - \mu}{\sigma} \quad \text{or} \quad -1.475 = \frac{35 - \mu}{\sigma}
\]
\[
z_2 = \frac{63 - \mu}{\sigma} \quad \text{or} \quad 1.227 = \frac{63 - \mu}{\sigma}
\]

### Step 3: Solve the system of equations.

We now solve these equations to find \( \mu \) and \( \sigma \). I'll compute this step next.The mean \( \mu \) of the distribution is approximately \( 50.28 \), and the standard deviation \( \sigma \) is approximately \( 10.36 \).

Would you like further details or clarifications on this solution?

Here are 5 related questions to explore further:
1. What is the probability of an item being above 70 in this distribution?
2. How do we find the z-scores for different probabilities?
3. What is the area under the normal curve between 35 and 63?
4. How do you interpret a z-score in a normal distribution?
5. What are the properties of the standard normal distribution?

**Tip**: In a standard normal distribution, z-scores represent the number of standard deviations a value is from the mean.

In a Normal Distribution 7% of the items are under 35 and 89% are under 63 determine the mean and standard deviation of the distribution

Determine the mean and standard deviation of a normal distribution where 7% of items are below 35 and 89% are below 63. This problem uses z-scores and the properties of normal distribution to solve. Suitable for students in Grades 10-12.

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