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This problem deals with the normal distribution of diastolic blood pressure levels in females. Let's break down the parts of the question:

- **Given data**: 
  - Mean diastolic blood pressure, \( \mu = 69.2 \) mm Hg.
  - Standard deviation, \( \sigma = 8 \) mm Hg.
  
### Part (a):
We are asked to find the percentage of females with diastolic blood pressure levels above 90 mm Hg, which is considered hypertensive. This requires calculating the probability that a randomly selected female has a blood pressure above 90 mm Hg.

This can be done by finding the corresponding z-score and then using the standard normal distribution table.

The z-score is calculated as:

\[
z = \frac{X - \mu}{\sigma}
\]
Where:
- \( X = 90 \) mm Hg (the threshold for hypertension),
- \( \mu = 69.2 \) mm Hg (mean),
- \( \sigma = 8 \) mm Hg (standard deviation).

### Part (b):
Here, we are asked to find the percentage of samples (with 16 females in each sample) where the sample mean exceeds 90 mm Hg. This is a problem of finding the probability that the sample mean is greater than 90 mm Hg. The distribution of the sample mean has a standard deviation known as the standard error, which is:

\[
SE = \frac{\sigma}{\sqrt{n}}
\]
Where:
- \( \sigma = 8 \) mm Hg,
- \( n = 16 \) (sample size).

Using the standard error, we can calculate a new z-score and find the percentage of samples where the sample mean exceeds 90 mm Hg.

I can now calculate both parts for you. Let's start with that.### Results:
- **Part (a)**: The percentage of females with diastolic blood pressure above 90 mm Hg (hypertension) is approximately **0.47%**.
- **Part (b)**: The percentage of samples with a mean diastolic blood pressure above 90 mm Hg is **0%**.

This means that, while a very small percentage of individual females may have hypertensive diastolic blood pressure, it is extremely unlikely that a sample of 16 females will have a mean pressure above 90 mm Hg.

Would you like more details or have any questions about the calculations?

Here are 5 related questions to further explore this topic:
1. How would the percentages change if the standard deviation was larger?
2. How is the standard error affected by increasing the sample size?
3. Can we use a different distribution to model blood pressure levels in different populations?
4. What percentage of females have a diastolic blood pressure between 70 and 80 mm Hg?
5. How would you calculate the probability for different thresholds of blood pressure?

**Tip**: When dealing with sample means, always remember that the standard error shrinks as the sample size increases, making extreme values less likely.

Diastolic blood pressure is a measure of the pressure when arteries rest between heartbeats. Suppose diastolic blood pressure levels in females are normally distributed with a mean of 69.2 mm Hg and a standard deviation of 8 mm Hg. Complete parts (a) and (b).
a. A diastolic blood pressure level above 90 mm Hg is considered to be hypertension. What percentage of females have hypertension?
b. If we randomly collect samples of females with 16 in each sample, what percentage of those samples have a mean above 90 mm Hg?

Learn how to calculate the percentage of females with diastolic blood pressure above 90 mm Hg and determine the likelihood of samples with a mean above 90 mm Hg. This problem involves the normal distribution, Z-scores, and the Central Limit Theorem. Suitable for high school students studying statistics (Grades 11-12).

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