Questions: Check your blood pressure: In a recent study, the Centers for Disease Control and Prevention reported that diastolic blood pressures of adult women in the United States are approximately normally distributed with mean 80.2 and standard deviation 9.6. (a) Find the 20th percentile of the blood pressures. (b) Find the first quartile of the blood pressures. Use the TI-84 Plus calculator and round the answers to at least two decimal places.

Solution Steps

The diastolic blood pressures of adult women in the United States are approximately normally distributed with the following parameters:

• Mean (\( \mu \)): \( 80.2 \)
• Standard Deviation (\( \sigma \)): \( 9.6 \)

To find the 20th percentile (\( P_{20} \)), we use the inverse of the cumulative distribution function (CDF) for a normal distribution:

\[ P_{20} = \mu + z_{0.20} \cdot \sigma \]

Using the Z-score for the 20th percentile, we find:

\[ P_{20} \approx 80.2 + (-0.8416) \cdot 9.6 \approx 72.12 \]

Thus, the 20th percentile of the blood pressures is:

\[ \boxed{P_{20} = 72.12} \]

To find the first quartile (\( Q_1 \)), which corresponds to the 25th percentile (\( P_{25} \)), we similarly use the inverse CDF:

\[ Q_1 = \mu + z_{0.25} \cdot \sigma \]

Using the Z-score for the 25th percentile, we find:

\[ Q_1 \approx 80.2 + (-0.6745) \cdot 9.6 \approx 73.72 \]

Thus, the first quartile of the blood pressures is:

\[ \boxed{Q_1 = 73.72} \]

Final Answer