3. Fifteen different medical students measure the blood pressure of the same person. The systolic readings (in mm Hg) are listed below. If the mean is 132.3 mmHg and the standard deviation is 10.3 mmHg, should any of the medical students retake the person’s blood pressure? In other words, are any of the readings significantly high or significantly low? Explain using the Range Rule of Thumb. [7points] 138 130 135 110 140 120 125 120 130 130 144 143 140 130 150

To determine if any of the systolic blood pressure readings are significantly high or low using the **Range Rule of Thumb**, we follow these steps:

### Step 1: Range Rule of Thumb
The **Range Rule of Thumb** states that most of the data values will fall within **2 standard deviations** of the mean. Therefore, we can define "significantly high" and "significantly low" values using this rule.

- **Low Boundary (Minimum Value) = Mean - 2 × Standard Deviation**
- **High Boundary (Maximum Value) = Mean + 2 × Standard Deviation**

### Step 2: Compute the Boundaries
Given:
- Mean \( \mu = 132.3 \) mmHg
- Standard deviation \( \sigma = 10.3 \) mmHg

Now, we calculate the boundaries:

- **Low Boundary**:
  \[
  \text{Low Boundary} = 132.3 - 2 \times 10.3 = 132.3 - 20.6 = 111.7 \, \text{mmHg}
  \]

- **High Boundary**:
  \[
  \text{High Boundary} = 132.3 + 2 \times 10.3 = 132.3 + 20.6 = 152.9 \, \text{mmHg}
  \]

### Step 3: Check the Systolic Readings
The systolic readings are:
\[ 138, 130, 135, 110, 140, 120, 125, 120, 130, 130, 144, 143, 140, 130, 150 \]

- The **Low Boundary** is \( 111.7 \) mmHg, so any value below this is considered significantly low.
- The **High Boundary** is \( 152.9 \) mmHg, so any value above this is considered significantly high.

Let’s check the values:
- **110 mmHg** is below the low boundary of \( 111.7 \) mmHg, so it is significantly low.
- All other values fall within the range \( [111.7, 152.9] \), so no other readings are significantly high or low.

### Step 4: Conclusion
According to the **Range Rule of Thumb**, only the reading of **110 mmHg** is significantly low. Therefore, the student who recorded the **110 mmHg** reading should retake the blood pressure measurement, as it is an outlier compared to the rest of the data.

Would you like more details or have any further questions?

### Related Questions:
1. How does the Range Rule of Thumb compare to other statistical methods for identifying outliers?
2. What factors can influence the variation in blood pressure readings between different measurements?
3. How would the results change if the standard deviation were larger or smaller?
4. Why is using standard deviation an effective method for detecting significantly high or low readings?
5. What are some potential sources of error in manual blood pressure measurement?

### Tip:
When analyzing data with many measurements, it's important to verify whether outliers are due to measurement error or real variability in the data.

Fifteen different medical students measure the blood pressure of the same person. The systolic readings (in mm Hg) are listed below. If the mean is 132.3 mmHg and the standard deviation is 10.3 mmHg, should any of the medical students retake the person’s blood pressure? In other words, are any of the readings significantly high or significantly low? Explain using the Range Rule of Thumb. 138 130 135 110 140 120 125 120 130 130 144 143 140 130 150

This problem explores how to identify significantly high or low blood pressure readings using the Range Rule of Thumb. With a mean of 132.3 mmHg and a standard deviation of 10.3 mmHg, find out if any of the 15 readings are outliers. Learn the key concepts of standard deviation and statistical boundaries in this detailed solution, ideal for Grades 10-12.

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