Im a normal distribution 31% of items are cinder 45 and 8%. are over 64. Find the mean and S.D of the distribution

To find the mean (\(\mu\)) and standard deviation (\(\sigma\)) of a normal distribution, given that 31% of items are under 45 and 8% are over 64, we can use the properties of the standard normal distribution (Z-distribution).

### Step 1: Convert percentages to Z-scores
- The 31% under 45 corresponds to a cumulative probability of 0.31.
- The 8% over 64 corresponds to a cumulative probability of 1 - 0.08 = 0.92.

Using Z-tables (or a statistical calculator), we can find the Z-scores corresponding to these probabilities.

1. **For the cumulative probability of 0.31**: The Z-score corresponding to 0.31 is approximately **-0.50**.
2. **For the cumulative probability of 0.92**: The Z-score corresponding to 0.92 is approximately **1.41**.

### Step 2: Set up the equations
Using the formula for the Z-score:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \(X\) is the value in the distribution, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

For \(X = 45\) (31% cumulative):
\[
-0.50 = \frac{45 - \mu}{\sigma} \quad \text{(Equation 1)}
\]

For \(X = 64\) (92% cumulative):
\[
1.41 = \frac{64 - \mu}{\sigma} \quad \text{(Equation 2)}
\]

### Step 3: Solve the equations
We now solve these two simultaneous equations for \(\mu\) (mean) and \(\sigma\) (standard deviation).

1. From Equation 1: 
   \[
   \mu = 45 + 0.50\sigma
   \]
   
2. From Equation 2:
   \[
   \mu = 64 - 1.41\sigma
   \]

Now, we equate the two expressions for \(\mu\):
\[
45 + 0.50\sigma = 64 - 1.41\sigma
\]

Solving for \(\sigma\):
\[
1.91\sigma = 64 - 45
\]
\[
1.91\sigma = 19
\]
\[
\sigma = \frac{19}{1.91} \approx 9.95
\]

### Step 4: Find \(\mu\)
Substituting \(\sigma \approx 9.95\) into Equation 1:
\[
\mu = 45 + 0.50 \times 9.95 \approx 49.98
\]

### Final Answer:
- The mean \(\mu\) is approximately **49.98**.
- The standard deviation \(\sigma\) is approximately **9.95**.

Would you like further details on any of these steps, or do you have any questions?

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Here are 5 related questions you might find interesting:

1. How do Z-scores relate to the standard normal distribution?
2. How can you find the cumulative probability using a Z-table?
3. What is the significance of the empirical rule (68-95-99.7 rule) in a normal distribution?
4. How do you calculate probabilities for non-standard normal distributions?
5. What are some real-life examples of normally distributed data?

**Tip:** When working with normal distributions, always check if the problem involves standard normal (mean = 0, standard deviation = 1) or another form, so you apply the right formulas and Z-scores.

Learn how to calculate the mean (\(\mu\)) and standard deviation (\(\sigma\)) of a normal distribution when given percentages of items under 45 and over 64. This problem involves using Z-scores and cumulative probabilities to derive the parameters of the distribution. Suitable for advanced high school students and those studying statistics.

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