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.