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.