Student Portal IMain MAT-274 Class Resources A) ALEKS - Jocelyn Chavez - Learn Random Variables and Distributions 15 Joc Normal distribution: Finding a probability, advanced A biology researcher is studying the risk of extinction of a rare tree species in a remote part of the Amazon. In the course of her study, the researcher models the trees' ages using a normal distribution with a mean of 245 years and a standard deviation of 50 years. Use this table or the ALEKS calculator to find the percentage of trees with an age less than 231 years or more than 259 years according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example \( 98.23 \% \) ). \( \square \) \%
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To solve this problem, we first find the z-scores for the ages of 231 years and 259 years using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] Where \( X \) is the value, \( \mu \) is the mean (245 years), and \( \sigma \) is the standard deviation (50 years). For 231 years: \[ z_{231} = \frac{(231 - 245)}{50} = -0.28 \] For 259 years: \[ z_{259} = \frac{(259 - 245)}{50} = 0.28 \] Next, using a standard normal distribution table or calculator, we find the probabilities for these z-scores. The cumulative probability for \( z = -0.28 \) is approximately \( 0.3897 \), and for \( z = 0.28 \) it is approximately \( 0.6103 \). Therefore, the probabilities of trees being less than 231 years or more than 259 years can be calculated as: \[ P(X < 231) = 0.3897 \] \[ P(X > 259) = 1 - 0.6103 = 0.3897 \] Adding these two probabilities gives: \[ P(X < 231 \text{ or } X > 259) = 0.3897 + 0.3897 = 0.7794 \] To convert this to a percentage, multiply by 100 to get approximately **77.94%**. Thus, the final answer rounded to two decimal places is **77.94%**.
