Step 1: Draw the Normal Distribution

The scenario involves a normal distribution centered around a mean of 71.5 mph, with a standard deviation of 3.5 mph. Here’s how we’ll structure the drawing:

1. Center Line for Mean: Start by marking the mean (71.5 mph) in the center of the x-axis.
2. Inflection Points and Standard Deviations: Mark one standard deviation (3.5 mph) from the mean in both directions. This represents the points where the curve starts bending away from its initial steep slope.
3. X-Axis Labels: Mark the locations of the mean $(\mu)$, and then show the points for $\mu \pm 1\sigma$, $\mu \pm 2\sigma$, and $\mu \pm 3\sigma$. No specific values are required, so just place these marks evenly on the x-axis.
4. Descriptive X-Axis Label: Label the x-axis as "Speed of Cheetah in mph."

I'll describe these elements here since I'm unable to hand-draw directly, but you can replicate this with a simple bell curve sketch.

Step 2: Calculate the Probability

To find the probability of a cheetah running faster than 75.5 mph, we need to determine how far this speed is from the mean in terms of standard deviations (z-scores) and then use the normal distribution.

1. Calculate the Z-Score: $Z = \frac{X - \mu}{\sigma} = \frac{75.5 - 71.5}{3.5} = \frac{4}{3.5} \approx 1.14$

2. Determine the Probability Using Z-Score: Using a Z-table or calculator, we find the area to the left of $Z = 1.14$, which gives the cumulative probability up to 75.5 mph.

   • The cumulative probability for $Z = 1.14$ is approximately $0.8729$.
   • Since we need the probability that a cheetah runs faster than 75.5 mph, we subtract this from 1: $P(X > 75.5) = 1 - 0.8729 = 0.1271$

   So, there’s about a 12.71% chance that a cheetah will run faster than 75.5 mph.

Step 3: Update the Hand-Drawn Picture

In your drawing:

• Mark 75.5 on the x-axis as it’s the threshold speed we’re analyzing.
• Add “hops” for the Z-score: Indicate the distance from the mean to 75.5 mph as 1.14 standard deviations.
• Shade the Region for Probability: Shade the area under the curve to the right of 75.5 mph to represent $P(X > 75.5)$.

Final Answer in Plain Language

"There’s about a 12.71% chance that an adult cheetah will run faster than 75.5 miles per hour."

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Would you like more clarification on any step or concept here?

Additional Questions

1. How would the probability change if the mean speed of cheetahs increased?
2. What if we wanted the probability that a cheetah would run slower than 68 mph?
3. How is the Z-score useful in comparing speeds across different distributions?
4. Can we find the exact speed range that contains 95% of all cheetah speeds?
5. What happens to the probability calculation if the standard deviation increases?

Tip: Z-scores are helpful for standardizing values, making it easier to calculate probabilities for any normal distribution.