It seems there is a small typo in the weight range you provided - you likely meant to say "229 and 259 pounds" instead of "229 and 25". I'll proceed with this assumption.
To find the probability that a randomly selected elephant from the herd weighs between 229 and 259 pounds, we can use the properties of the normal distribution, given that the mean weight (μ) is 244 pounds and the standard deviation (σ) is 15 pounds.
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Standardize the weights: We will convert the weights to z-scores using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \(X\) is the value of interest.
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Calculate the z-scores for 229 and 259 pounds:
- For 229 pounds: \[ z_{229} = \frac{(229 - 244)}{15} = \frac{-15}{15} = -1 \]
- For 259 pounds: \[ z_{259} = \frac{(259 - 244)}{15} = \frac{15}{15} = 1 \]
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Find the probabilities associated with these z-scores using the standard normal distribution table (or a calculator):
- \(P(Z < -1)\) corresponds to the z-score of -1, which is approximately 0.1587.
- \(P(Z < 1)\) corresponds to the z-score of 1, which is approximately 0.8413.
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Calculate the probability that the weight is between 229 and 259 pounds: \[ P(229 < X < 259) = P(Z < 1) - P(Z < -1) \] \[ P(229 < X < 259) = 0.8413 - 0.1587 = 0.6826 \]
Thus, the probability that a randomly selected elephant will weigh between 229 and 259 pounds is approximately 0.6826 or 68.26%.