Given that the systolic blood pressure is normally distributed with:
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Question 2
Given that the systolic blood pressure is normally distributed with:
Mean ($\mu$) = 94 mmHg
Variance ($\sigma^2$) = 121 mmHg
Standard deviation ($\sigma$) = $\sqrt{121} = 11$ mmHg
(a) What is the probability of a 5-year-old boy having a systolic blood pressure less than 70 mmHg?
Step 1: Calculate the z-score for $x = 70$.
$$ z = \frac{x - \mu}{\sigma} = \frac{70 - 94}{11} = \frac{-24}{11} \approx -2.18 $$
Step 2: Find the probability $P(X < 70)$ using the standard normal distribution table.
$$ P(X < 70) = P(Z < -2.18) $$
From the Z-table, $P(Z < -2.18) \approx 0.0146$.
The probability is $\boxed{0.0146}$.
(b) What is the probability of a 5-year-old boy having a systolic blood pressure higher than 100 mmHg?
Step 1: Calculate the z-score for $x = 100$.
$$ z = \frac{x - \mu}{\sigma} = \frac{100 - 94}{11} = \frac{6}{11} \approx 0.55 $$
Step 2: Find the probability $P(X > 100)$ using the standard normal distribution table.
$$ P(X > 100) = P(Z > 0.55) = 1 - P(Z \le 0.55) $$
From the Z-table, $P(Z \le 0.55) \approx 0.7088$.
$$ P(Z > 0.55) = 1 - 0.7088 = 0.2912 $$
The probability is $\boxed{0.2912}$.
(c) What is the probability that the blood pressure of a 5-year-old boy will be between 80 and 100 mmHg?
Step 1: Calculate the z-score for $x_1 = 80$.
$$ z_1 = \frac{80 - 94}{11} = \frac{-14}{11} \approx -1.27 $$
Step 2: Calculate the z-score for $x_2 = 100$.
$$ z_2 = \frac{100 - 94}{11} = \frac{6}{11} \approx 0.55 $$
Step 3: Find the probability $P(80 < X < 100) = P(-1.27 < Z < 0.55)$.
$$ P(-1.27 < Z < 0.55) = P(Z < 0.55) - P(Z < -1.27) $$
From the Z-table:
$P(Z < 0.55) \approx 0.7088$
$P(Z < -1.27) \approx 0.1020$
$$ P(-1.27 < Z < 0.55) = 0.7088 - 0.1020 = 0.6068 $$
The probability is $\boxed{0.6068}$.
(d) What systolic blood pressure is necessary to put someone in the bottom 10% of the distribution?
Step 1: Find the z-score corresponding to the bottom 10% (i.e., $P(Z < z) = 0.10$).
From the Z-table, the z-score that corresponds to a cumulative probability of 0.10 is approximately $z = -1.28$.
Step 2: Use the z-score formula to solve for $x$.
$$ z = \frac{x - \mu}{\sigma} \implies x = \mu + z\sigma $$
$$ x = 94 + (-1.28)(11) $$
$$ x = 94 - 14.08 $$
$$ x = 79.92 $$
The systolic blood pressure is approximately $\boxed{79.92 \text{ mmHg}}$.
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