The mean IQ score for 1400 students is 100 , with a standard deviation of 5 . Assuming the scores have a normal curve, answer the following a. How many have an IQ score between 95 and 105 ? b. How many have an IQ score between 90 and 110 ? c. How many have an IQ score over 115 ? a. The number of students with an IQ score between 95 and 105 is 956 . (Round to the nearest person as needed.)

Let's analyze each part of the problem using the properties of the normal distribution. **Given:** - **Mean (μ)** = 100 - **Standard Deviation (σ)** = 5 - **Number of Students (N)** = 1400 ### a. Number of students with IQ between 95 and 105 You've already provided the answer: - **Answer:** 956 students This is correct. Approximately 68% of data in a normal distribution lies within ±1 standard deviation from the mean. - Calculation: 0.68 × 1400 ≈ 952 (rounded to 956 for more precise computation). --- ### b. Number of students with IQ between 90 and 110 **Range:** 90 to 110 1. **Calculate Z-scores:** - \( Z_1 = \frac{90 - 100}{5} = -2 \) - \( Z_2 = \frac{110 - 100}{5} = 2 \) 2. **Find the Probability Between Z = -2 and Z = 2:** - This range corresponds to approximately 95.45% of the data in a normal distribution. 3. **Calculate the Number of Students:** - \( 0.9545 \times 1400 \approx 1336 \) **Answer:** Approximately **1,336 students** have an IQ between 90 and 110. --- ### c. Number of students with IQ over 115 **Range:** Above 115 1. **Calculate Z-score:** - \( Z = \frac{115 - 100}{5} = 3 \) 2. **Find the Probability Above Z = 3:** - \( P(Z > 3) \) ≈ 0.00135 (or 0.135%) 3. **Calculate the Number of Students:** - \( 0.00135 \times 1400 \approx 1.89 \) 4. **Round to the Nearest Whole Number:** - Approximately **2 students** **Answer:** Approximately **2 students** have an IQ over 115. --- ### Summary of Answers: - **a.** 956 students - **b.** 1,336 students - **c.** 2 students