A researcher studying frogs is investigating the distance that a certain species of frog can jump. The jump lengths appear to be approximately normally distributed with a mean of 90 inches and a standard deviation of 8 inches. Directions: Use the "empirical rule" \( (68-95-99.7) \) for this setting. a) What proportion of frog jumps are less than 74 inches? b) What jump lengths represent the middle \( 95 \% \) of frog jumps? Between 74 106 c) What is the probability of observing a random frog jump that is longer than 98 inches? 16 Submit Question

Let's analyze each part of the problem using the **empirical rule** (also known as the 68-95-99.7 rule) for normally distributed data. **Given:** - **Mean (μ):** 90 inches - **Standard Deviation (σ):** 8 inches The empirical rule states: - **68%** of the data falls within **±1σ** (82 to 98 inches). - **95%** of the data falls within **±2σ** (74 to 106 inches). - **99.7%** of the data falls within **±3σ** (66 to 114 inches). --- ### a) **Proportion of Frog Jumps Less Than 74 Inches** 74 inches is exactly **2 standard deviations below the mean** (μ - 2σ = 90 - 16 = 74 inches). - According to the empirical rule, **95%** of the data lies between 74 and 106 inches. - This leaves **5%** of the data in the two tails (lower than 74 inches and higher than 106 inches). - Since the distribution is symmetric, **2.5%** of the frog jumps are **less than 74 inches**. **Answer:** Approximately **2.5%** of frog jumps are less than 74 inches. --- ### b) **Jump Lengths Representing the Middle 95% of Frog Jumps** - The middle **95%** of the data falls within **±2σ** from the mean. - Calculated as: - Lower bound: μ - 2σ = 90 - 16 = **74 inches** - Upper bound: μ + 2σ = 90 + 16 = **106 inches** **Answer:** The middle **95%** of frog jumps are between **74 inches and 106 inches**. --- ### c) **Probability of a Frog Jump Longer Than 98 Inches** 98 inches is exactly **1 standard deviation above the mean** (μ + σ = 90 + 8 = 98 inches). - According to the empirical rule: - **68%** of the data lies between 82 inches and 98 inches. - This leaves **32%** of the data outside this range. - Since the distribution is symmetric, **16%** of the jumps are **longer than 98 inches**. **Answer:** There is a **16%** probability of observing a frog jump longer than 98 inches.