Pregunta
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\( X \) is a normally distributed random variable with mean 50 and standard deviation 10 . What is the probability that \( X \) is less than 30 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

Ask by Guzman Medina. in the United States
Jan 24,2025

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The probability that \( X \) is less than 30 is 0.025.

Solución

To find the probability that \( X \) is less than 30, where \( X \) is normally distributed with a mean (\( \mu \)) of 50 and a standard deviation (\( \sigma \)) of 10, we can use the \( 68-95-99.7 \) rule (also known as the empirical rule). 1. **Identify the number of standard deviations from the mean:** \[ 30 = \mu - 2\sigma = 50 - 2(10) = 50 - 20 = 30 \] So, 30 is 2 standard deviations below the mean. 2. **Apply the \( 68-95-99.7 \) rule:** - Approximately **68%** of the data lies within **±1** standard deviation from the mean (40 to 60). - Approximately **95%** of the data lies within **±2** standard deviations from the mean (30 to 70). - Approximately **99.7%** of the data lies within **±3** standard deviations from the mean (20 to 80). 3. **Calculate the probability:** - Since 95% of the data lies between 30 and 70, the remaining 5% is split equally in the two tails of the distribution. - Therefore, the probability that \( X \) is less than 30 is: \[ P(X < 30) = \frac{5\%}{2} = 2.5\% = 0.025 \] **Final Answer:** The probability is 0.025.

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The Deep Dive

To find the probability that \( X \) is less than 30 using the \( 0.68-0.95-0.997 \) rule (empirical rule for normal distributions), we first calculate how many standard deviations \( 30 \) is from the mean. The mean is \( 50 \) and the standard deviation is \( 10 \). We calculate the z-score: \[ z = \frac{X - \text{mean}}{\text{standard deviation}} = \frac{30 - 50}{10} = \frac{-20}{10} = -2 \] A z-score of -2 indicates that \( 30 \) is 2 standard deviations below the mean. According to the empirical rule, about 95% of the data falls within 2 standard deviations of the mean, meaning that 2.5% is below \( \mu - 2\sigma \) (which is 30 in this case). Therefore, the probability that \( X \) is less than 30 is approximately \[ P(X < 30) ≈ 0.025 \] So the answer, rounded to the nearest thousandth, is: 0.025

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