A mammalogist models the daily movement distance of a baboon as a normal distribution with mean 8 km and standard deviation 1.2 km. What is the approximate probability that a randomly selected day’s distance exceeds 10 km?

["Approximate Probability a Baboon Travels More Than 10 km in a Day: A Mammalogist’s Model Using Normal Distribution", "In behavioral ecology, understanding the spatial movement of animals provides vital insights into their foraging patterns, social interactions, and habitat use. A recent study by a mammalogist models the daily movement distance of a baboon as a normal distribution with a mean of 8 kilometers and a standard deviation of 1.2 kilometers. One key question is: what is the probability that a randomly selected day’s travel distance exceeds 10 km?", "### Understanding the Normal Distribution Model\nThe mammalogist assumes the daily movement distance ( X ) follows:\n[ X \sim N(\mu = 8,\ \sigma = 1.2) ]\nThis means most baboons travel near 8 km per day, with daily distances spread around that central value by ±1.2 km or more, depending on variability.", "### Calculating the Z-Score\nTo find ( P(X > 10) ), we first convert the raw score to a Z-score using the formula:\n[\nZ = \frac{X - \mu}{\sigma} = \frac{10 - 8}{1.2} = \frac{2}{1.2} \approx 1.67\n]\nThis Z-score of approximately 1.67 indicates how many standard deviations 10 km lies above the mean.", "### Finding the Probability Using Standard Normal Table\nWe now find the area under the standard normal curve to the right of ( Z = 1.67 ). From standard normal distribution tables:\n- The cumulative probability ( P(Z \leq 1.67) \approx 0.9525 )\n- Therefore,\n[\nP(X > 10) = P(Z > 1.67) = 1 - 0.9525 = 0.0475\n]\nThis corresponds to roughly 4.75%.", "### Interpretation and Ecological Insight\nThe result shows that on about 4.75% of days, a baboon is likely to travel more than 10 km—nearly 2.3 times the average. Such long-distance movements might occur during resource scarcity, territorial patrols, or exploratory behaviors, highlighting the flexibility in baboon spatial strategies.", "### Summary\n- Mean movement: 8 km\n- Standard deviation: 1.2 km\n- 10 km distance ≈ 1.67 standard deviations above the mean\n- Probability of exceeding 10 km: ≈ 4.75%", "This model exemplifies how statistical normalization enhances understanding of animal behavior, enabling researchers to predict and interpret movement patterns within ecological contexts.", "---", "Key terms: mammalogist, baboon movement, normal distribution, mean (8 km), standard deviation (1.2 km), Z-score, probability calculation, behavioral ecology"]