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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You're on a roll — Question 2 Given that the systolic blood pressure is normally distributed with: Mean () = 94 mmHg Variance (^2) = 121 mmHg Standard deviation () = 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 = (x - )/() = (70 - 94)/(11) = (-24)/(11) ≈ -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) ≈ 0.0146. The probability is 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 = (x - )/() = (100 - 94)/(11) = (6)/(11) ≈ 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 0.55) From the Z-table, P(Z 0.55) ≈ 0.7088. P(Z > 0.55) = 1 - 0.7088 = 0.2912 The probability is 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 = (80 - 94)/(11) = (-14)/(11) ≈ -1.27 Step 2: Calculate the z-score for x_2 = 100. z_2 = (100 - 94)/(11) = (6)/(11) ≈ 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) ≈ 0.7088 P(Z < -1.27) ≈ 0.1020 P(-1.27 < Z < 0.55) = 0.7088 - 0.1020 = 0.6068 The probability is 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 = (x - )/() x = + z x = 94 + (-1.28)(11) x = 94 - 14.08 x = 79.92 The systolic blood pressure is approximately 79.92 mmHg. What's next? Send 'em!