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To determine the percentage of employees with IQs between 90 and 110, we will need to calculate the area under the standard normal curve for the corresponding \( z \)-scores.

### Step 1: Calculate the \( z \)-scores
Given that the IQ distribution is normal with a mean of 95 and a standard deviation of 12.5, we can compute the \( z \)-scores for IQs 90 and 110 using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \( X \) is the IQ score,
- \( \mu = 95 \) is the mean,
- \( \sigma = 12.5 \) is the standard deviation.

#### For IQ = 90:
\[
z_1 = \frac{90 - 95}{12.5} = \frac{-5}{12.5} = -0.4
\]

#### For IQ = 110:
\[
z_2 = \frac{110 - 95}{12.5} = \frac{15}{12.5} = 1.2
\]

### Step 2: Look up the cumulative probabilities
Using the standard normal table:
- For \( z_1 = -0.4 \), we look up the value for \( z = 0.4 \) (since the table is symmetric), which gives a probability of 0.3446.
- For \( z_2 = 1.2 \), the table gives a probability of 0.8849.

### Step 3: Calculate the percentage
The percentage of employees with IQs between 90 and 110 is the difference between these two probabilities:
\[
P(90 < IQ < 110) = 0.8849 - 0.3446 = 0.5403
\]
So, the percentage is approximately \( 54.03\% \).

Would you like any more details or have further questions? 

Here are 5 related questions for further practice:
1. What is the probability of an employee having an IQ above 110?
2. How would the percentage change if the standard deviation was 10 instead of 12.5?
3. What is the percentage of employees with IQs below 90?
4. How do we find the probability of an employee having an IQ between 100 and 120?
5. How do we interpret \( z \)-scores in terms of standard deviations from the mean?

**Tip:** The standard normal distribution is symmetrical, which means you can easily calculate probabilities for negative \( z \)-scores by using the positive ones from the table.

A large company employs workers whose IQs are distributed normally with a mean of 95 and standard deviation 12.5. What percent of employees would have IQs between 90 and 110?

Learn how to calculate the percentage of employees with IQs between 90 and 110, given a normal distribution with a mean of 95 and standard deviation of 12.5. This problem involves calculating Z-scores and finding cumulative probabilities using the standard normal table.

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