Questions: The systolic blood pressure of adults in the USA is nearly normally distributed with a mean of 120 millimeters of mercury ( mmHg ) and standard deviation of 26. Someone qualifies as having Stage 2 high blood pressure if their systolic blood pressure is 160 or higher. Stage 1 high BP is specified as systolic BP between 140 and 160. Give your answers rounded to 4 decimal places. a. What is the probability that an adult in the USA has stage 2 high blood pressure? b. What is the probability that an adult in the USA has stage 1 high blood pressure? c. Your doctor tells you you are in the 30th percentile for blood pressure among US adults. What is your systolic BP? mmHg d. What is the systolic blood pressure that cuts off the top 2.5% of adults in the USA? mmHg

The systolic blood pressure of adults in the USA is nearly normally distributed with a mean of 120 millimeters of mercury ( mmHg ) and standard deviation of 26. Someone qualifies as having Stage 2 high blood pressure if their systolic blood pressure is 160 or higher. Stage 1 high BP is specified as systolic BP between 140 and 160. Give your answers rounded to 4 decimal places. a. What is the probability that an adult in the USA has stage 2 high blood pressure? b. What is the probability that an adult in the USA has stage 1 high blood pressure? c. Your doctor tells you you are in the 30th percentile for blood pressure among US adults. What is your systolic BP? mmHg d. What is the systolic blood pressure that cuts off the top 2.5% of adults in the USA? mmHg
Transcript text: The systolic blood pressure of adults in the USA is nearly normally distributed with a mean of 120 millimeters of mercury ( mmHg ) and standard deviation of 26. Someone qualifies as having Stage 2 high blood pressure if their systolic blood pressure is 160 or higher. Stage 1 high BP is specified as systolic BP between 140 and 160. Give your answers rounded to 4 decimal places. a. What is the probability that an adult in the USA has stage 2 high blood pressure? $\square$ b. What is the probability that an adult in the USA has stage 1 high blood pressure? $\square$ c. Your doctor tells you you are in the 30th percentile for blood pressure among US adults. What is your systolic BP? $\square$ mmHg d. What is the systolic blood pressure that cuts off the top $2.5 \%$ of adults in the USA? $\square$ mmHg
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Solution

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Solution Steps

Step 1: Probability of Stage 2 High Blood Pressure

To find the probability that an adult in the USA has Stage 2 high blood pressure (systolic BP \( \geq 160 \) mmHg), we calculate:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(1.5385) = 0.062 \]

Thus, the probability of Stage 2 high blood pressure is:

\[ \boxed{P = 0.062} \]

Step 2: Probability of Stage 1 High Blood Pressure

Next, we determine the probability that an adult has Stage 1 high blood pressure (systolic BP between \( 140 \) and \( 160 \) mmHg):

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.5385) - \Phi(0.7692) = 0.1589 \]

Therefore, the probability of Stage 1 high blood pressure is:

\[ \boxed{P = 0.1589} \]

Step 3: Systolic Blood Pressure at the 30th Percentile

To find the systolic blood pressure corresponding to the 30th percentile, we use the inverse of the cumulative distribution function:

\[ \text{Systolic BP} = 106.3656 \text{ mmHg} \]

Thus, the systolic blood pressure at the 30th percentile is:

\[ \boxed{BP = 106.3656 \text{ mmHg}} \]

Final Answer

  • Probability of Stage 2 high blood pressure: \( \boxed{0.062} \)
  • Probability of Stage 1 high blood pressure: \( \boxed{0.1589} \)
  • Systolic BP at the 30th percentile: \( \boxed{106.3656 \text{ mmHg}} \)
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