Questions: 18. Blood pressure According to a health information website, the distribution of adults' diastolic blood pressure (in millimeters of mercury, mmHg ) can be modeled by a normal distribution with mean 70 mmHg and standard deviation 20 mmHg . A diastolic pressure greater than 100 mmHg for an adult is classified as very high blood pressure. About what proportion of adults have very high blood pressure according to this criterion?

Solution Steps

To determine the Z-score for a diastolic blood pressure of \(100\) mmHg, we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

where:

• \(X = 100\) mmHg (the threshold for very high blood pressure),
• \(\mu = 70\) mmHg (the mean),
• \(\sigma = 20\) mmHg (the standard deviation).

Substituting the values, we have:

\[ z = \frac{100 - 70}{20} = \frac{30}{20} = 1.5 \]

Thus, the Z-score for the threshold \(100\) mmHg is \(z = 1.5\).

Next, we need to find the probability of an adult having a diastolic blood pressure greater than \(100\) mmHg. This can be expressed as:

\[ P(X > 100) = P(Z > 1.5) = \Phi(\infty) - \Phi(1.5) \]

Using the cumulative distribution function \( \Phi \), we find:

\[ P(X > 100) = 0.0668 \]

This indicates that approximately \(6.68\%\) of adults have very high blood pressure.

Final Answer