In a recent study, the Centers for Disease Control and Prevention reported that diastolic blood pressures (inmmHg) of adult women in the United States are approximately normally distributed with mean 80.4 and standard deviation 9.5.

(a) Find the 20th percentile of the blood pressures.

(b) Find the 60th percentile of the blood pressures.

(c) Find the third quartile of the blood pressures.

Use Excel and round the answers to at least two decimal places.

The problem requires the use of the normal distribution to find percentiles. The mean (\(\mu\)) of the diastolic blood pressures is 80.4, and the standard deviation (\(\sigma\)) is 9.5. For each part, we will find the Z-score corresponding to the desired percentile and then use the following formula to find the actual value of the blood pressure at that percentile:

\[
X = \mu + Z \cdot \sigma
\]

Where:
- \(X\) is the desired value (blood pressure),
- \(\mu = 80.4\) is the mean,
- \(\sigma = 9.5\) is the standard deviation,
- \(Z\) is the Z-score corresponding to the given percentile.

Let's walk through the steps for each part.

### (a) Find the 20th percentile of the blood pressures

1. **Find the Z-score for the 20th percentile.** 
   - Use the Excel function `NORM.S.INV(0.20)` to find the Z-score.
   - The Z-score for the 20th percentile is approximately \(-0.8416\).

2. **Calculate the blood pressure value:**
   \[
   X = 80.4 + (-0.8416) \cdot 9.5 = 80.4 - 7.9952 = 72.40
   \]

So, the 20th percentile of blood pressures is approximately **72.40 mmHg**.

---

### (b) Find the 60th percentile of the blood pressures

1. **Find the Z-score for the 60th percentile.** 
   - Use the Excel function `NORM.S.INV(0.60)` to find the Z-score.
   - The Z-score for the 60th percentile is approximately **0.2533**.

2. **Calculate the blood pressure value:**
   \[
   X = 80.4 + (0.2533) \cdot 9.5 = 80.4 + 2.40635 = 82.81
   \]

So, the 60th percentile of blood pressures is approximately **82.81 mmHg**.

---

### (c) Find the third quartile (75th percentile) of the blood pressures

1. **Find the Z-score for the 75th percentile.** 
   - Use the Excel function `NORM.S.INV(0.75)` to find the Z-score.
   - The Z-score for the 75th percentile is approximately **0.6745**.

2. **Calculate the blood pressure value:**
   \[
   X = 80.4 + (0.6745) \cdot 9.5 = 80.4 + 6.40775 = 86.81
   \]

So, the third quartile of blood pressures is approximately **86.81 mmHg**.

---

### Final Answers:
- (a) 20th percentile: **72.40 mmHg**
- (b) 60th percentile: **82.81 mmHg**
- (c) 75th percentile: **86.81 mmHg**

Would you like any further clarification or additional details?

Here are five related questions:
1. How do you interpret Z-scores in a normal distribution?
2. What is the significance of the third quartile in a data set?
3. How can you calculate the 95th percentile for this dataset?
4. How can these calculations be used in medical research?
5. How does the standard deviation affect the distribution of percentiles?

**Tip:** Excel’s `NORM.INV` function can directly calculate percentiles when provided with a mean and standard deviation.

In a recent study, the Centers for Disease Control and Prevention reported that diastolic blood pressures (in mmHg) of adult women in the United States are approximately normally distributed with mean 80.4 and standard deviation 9.5.

(a) Find the 20th percentile of the blood pressures.

(b) Find the 60th percentile of the blood pressures.

(c) Find the third quartile of the blood pressures.

Learn how to calculate the 20th, 60th percentiles, and the third quartile (75th percentile) of diastolic blood pressures in adult women using the normal distribution formula. The problem provides a mean of 80.4 mmHg and a standard deviation of 9.5 mmHg. This solution walks through step-by-step calculations using Z-scores, suitable for students in Grades 11-12.

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