The postoperative quality of life (QOL) scores of 200 prostate cancer patients have a mean of 60 and a standard detion of 10. How many patients are expected to have a QOL score between 40 and 80?
**Core Concept**
The core concept here is understanding the normal distribution of a population's mean and standard deviation, and using it to estimate the number of individuals within a specified range. In this case, we're dealing with a normal distribution of postoperative quality of life (QOL) scores in prostate cancer patients.
**Why the Correct Answer is Right**
To solve this problem, we'll use the 68-95-99.7 rule, also known as the empirical rule. This rule states that about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations. Since we're interested in the range between 40 and 80, we'll calculate the z-scores for these values. The z-score for 40 is (40 - 60) / 10 = -2, and the z-score for 80 is (80 - 60) / 10 = 2. Using a standard normal distribution table or calculator, we find that about 95.5% of the data falls between z-scores of -2 and 2. Therefore, we can expect about 95.5% of the patients to have a QOL score between 40 and 80.
**Why Each Wrong Option is Incorrect**
**Option A:** This option is likely incorrect because it's a very low percentage, and the 68-95-99.7 rule suggests that a much higher percentage of patients should fall within the specified range.
**Option B:** This option is likely incorrect because it's a very high percentage, and while it's true that about 95% of patients should fall within two standard deviations of the mean, the specified range goes beyond two standard deviations.
**Option D:** This option is likely incorrect because it's a very low percentage, and as we calculated earlier, about 95.5% of patients should fall within the specified range.
**Clinical Pearl / High-Yield Fact**
When dealing with normal distributions, the 68-95-99.7 rule is a useful tool for estimating the number of individuals within a specified range. However, be aware that this rule assumes a normal distribution, and real-world data may not always be perfectly normal.
**Correct Answer:** **C**. About 95.5% of the patients are expected to have a QOL score between 40 and 80.