A hospital wants to determine the appropriate dosage of a medication for a certain condition. The effectiveness of the medication is known to follow a normal distribution with a mean dosage of 150 mg and a standard deviation of 20 mg. What percentage of patients should receive a dosage between 130 mg and 170 mg for the medication to be effective?

Solution:

To solve this problem, we need to determine the percentage of patients who should receive a dosage between 130 mg and 170 mg, given that the effectiveness of the medication follows a **normal distribution** with a **mean (μ)** of 150 mg and a **standard deviation (σ)** of 20 mg.

### Step 1: Standardize the Dosages
We will convert the dosages of 130 mg and 170 mg into **z-scores** using the formula:

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

Where:
- \( x \) is the given dosage,
- \( \mu \) is the mean (150 mg),
- \( \sigma \) is the standard deviation (20 mg).

#### For 130 mg:
\[
z_{130} = \frac{130 - 150}{20} = \frac{-20}{20} = -1
\]

#### For 170 mg:
\[
z_{170} = \frac{170 - 150}{20} = \frac{20}{20} = 1
\]

### Step 2: Use the Z-Table (or Normal Distribution Table)
We now need to find the probability associated with each z-score from the **standard normal distribution**.

- For \( z = -1 \), the cumulative probability is approximately **0.1587**.
- For \( z = 1 \), the cumulative probability is approximately **0.8413**.

### Step 3: Calculate the Percentage of Patients
To find the percentage of patients who should receive a dosage between 130 mg and 170 mg, subtract the probability for \( z = -1 \) from the probability for \( z = 1 \):

\[
P(130 \, \text{mg} < x < 170 \, \text{mg}) = P(z = 1) - P(z = -1)
\]
\[
P(130 \, \text{mg} < x < 170 \, \text{mg}) = 0.8413 - 0.1587 = 0.6826
\]

### Step 4: Convert to Percentage
To express this probability as a percentage, multiply by 100:

\[
0.6826 \times 100 = 68.26\%
\]

### Final Answer:
**Approximately 68.26% of patients should receive a dosage between 130 mg and 170 mg for the medication to be effective.**

Would you like further details on this solution?

Here are 5 related questions to consider:
1. How do you calculate the z-score for a given value in a normal distribution?
2. What is the significance of the standard normal distribution in medicine?
3. How can this dosage range be adjusted if the standard deviation changes?
4. What percentage of patients would need a dosage less than 130 mg?
5. How can we interpret the results if the dosage range is extended to 120 mg to 180 mg?

**Tip:** Always visualize the normal distribution curve when working with probabilities to better understand the areas representing certain ranges.

A hospital wants to determine the appropriate dosage of a medication for a certain condition. The effectiveness of the medication is known to follow a normal distribution with a mean dosage of 150 mg and a standard deviation of 20 mg. What percentage of patients should receive a dosage between 130 mg and 170 mg for the medication to be effective?

Discover how to calculate the percentage of patients who should receive a dosage between 130 mg and 170 mg using the normal distribution with a mean of 150 mg and a standard deviation of 20 mg. This solution involves z-score calculations and is suitable for students in Grades 10-12.

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