For the purposes of constructing modified boxplots, outliers are defined as data values that are above \( \mathrm{Q}_{3} \) by an amount greater than \( 1.5 \times 1 \mathrm{QR} \) or below \( \mathrm{Q}_{1} \) by an amoun interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier. The probability that a randomly selected value taken from a normal distribution is considered an outlier is (Round to four decimal places as needed.)

To find the probability that a value randomly selected from a normal distribution is an outlier using the modified boxplot definition, we first calculate the thresholds for outliers. 1. Identify \( Q_1 \) (the first quartile) and \( Q_3 \) (the third quartile): In a normal distribution: - \( Q_1 \approx \mu - 0.675\sigma \) - \( Q_3 \approx \mu + 0.675\sigma \) 2. Calculate the interquartile range (IQR): \[ IQR = Q_3 - Q_1 \approx \mu + 0.675\sigma - (\mu - 0.675\sigma) = 1.35\sigma \] 3. Define the boundaries for outliers: - Upper boundary: \( Q_3 + 1.5 \times IQR = \mu + 0.675\sigma + 1.5 \times 1.35\sigma = \mu + 0.675\sigma + 2.025\sigma = \mu + 2.7\sigma \) - Lower boundary: \( Q_1 - 1.5 \times IQR = \mu - 0.675\sigma - 1.5 \times 1.35\sigma = \mu - 0.675\sigma - 2.025\sigma = \mu - 2.7\sigma \) 4. Using the normal distribution properties, find the probabilities: - The probability of being greater than \( \mu + 2.7\sigma \) and the probability of being less than \( \mu - 2.7\sigma \) can be found using the standard Z-scores. - For \( Z > 2.7 \), using a standard normal distribution table or calculator, \( P(Z > 2.7) \approx 0.0035 \). - For \( Z < -2.7 \), it is also \( P(Z < -2.7) \approx 0.0035 \). 5. Combine these probabilities: \[ P(\text{outlier}) = P(Z > 2.7) + P(Z < -2.7) \approx 0.0035 + 0.0035 = 0.0070 \] So, the probability that a randomly selected value from a normal distribution is considered an outlier is approximately \( \boxed{0.0070} \).