. 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 150mg 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 determine what percentage of patients should receive a dosage between 130 mg and 170 mg, given that the effectiveness of the medication follows a normal distribution with a mean of 150 mg and a standard deviation of 20 mg, we can use the properties of the **normal distribution**.

### Step-by-Step Solution:

1. **Identify the parameters of the normal distribution**:
   - Mean (\(\mu\)) = 150 mg
   - Standard deviation (\(\sigma\)) = 20 mg

2. **Standardize the dosage values (convert to z-scores)**:
   We need to find the **z-scores** corresponding to the dosages of 130 mg and 170 mg. The formula to convert a value \(X\) to a z-score is:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   
   For \(X = 130\) mg:
   \[
   Z_{130} = \frac{130 - 150}{20} = \frac{-20}{20} = -1
   \]
   
   For \(X = 170\) mg:
   \[
   Z_{170} = \frac{170 - 150}{20} = \frac{20}{20} = 1
   \]

3. **Use the standard normal distribution table** (or calculator) to find the probability corresponding to these z-scores:
   
   - The z-score of -1 corresponds to a cumulative probability of approximately 0.1587.
   - The z-score of 1 corresponds to a cumulative probability of approximately 0.8413.

4. **Calculate the percentage between these two z-scores**:
   The percentage of patients who should receive a dosage between 130 mg and 170 mg is the difference between the cumulative probabilities at the two z-scores:
   \[
   P(130 \leq X \leq 170) = P(Z \leq 1) - P(Z \leq -1) = 0.8413 - 0.1587 = 0.6826
   \]

5. **Convert the result to a percentage**:
   \[
   0.6826 \times 100 = 68.26\%
   \]

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

---

Do you have any questions or need further details on the calculation? Here are some related questions:

1. How do you calculate z-scores for values outside the range of the standard normal distribution?
2. What is the significance of the mean and standard deviation in a normal distribution?
3. How would the percentage change if the standard deviation was 15 mg instead of 20 mg?
4. How can we determine the dosage range for 95% of the population?
5. Can we apply the same method if the distribution is not normal?

**Tip**: In a normal distribution, about 68% of the data falls within one standard deviation of the mean. This is useful for quick approximations.

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 150mg 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?

Learn how to calculate the percentage of patients receiving an effective dosage between 130 mg and 170 mg, using normal distribution with a mean of 150 mg and standard deviation of 20 mg. This step-by-step guide covers the use of z-scores, cumulative probabilities, and statistical formulas to determine the effectiveness of the medication. Suitable for college-level statistics or advanced high school students.

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