\( 1 \leftarrow \) 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 ? 1 c. How many have an IQ score over 115 ? a. The number of students with an IQ score between 95 and 105 is (Round to the nearest person as needed.)

To find the number of students with an IQ score between 95 and 105, we can use the properties of the normal distribution. The mean \( \mu \) is 100 and the standard deviation \( \sigma \) is 5. Now, we will calculate the z-scores for 95 and 105: For 95: \[ z = \frac{(95 - 100)}{5} = -1 \] For 105: \[ z = \frac{(105 - 100)}{5} = 1 \] Using the standard normal distribution table, we find: - The area below \( z = 1 \) is approximately 0.8413. - The area below \( z = -1 \) is approximately 0.1587. To find the area between these z-scores: \[ 0.8413 - 0.1587 = 0.6826 \] Now, multiply by the total number of students: \[ 0.6826 \times 1400 \approx 956 \] So, the number of students with an IQ score between 95 and 105 is approximately **956 students**. Now, regarding your other queries: For part b, to find how many students have an IQ score between 90 and 110: Calculate the z-scores for 90 and 110: For 90: \[ z = \frac{(90 - 100)}{5} = -2 \] For 110: \[ z = \frac{(110 - 100)}{5} = 2 \] Using the standard normal distribution table again: - The area below \( z = 2 \) is approximately 0.9772. - The area below \( z = -2 \) is approximately 0.0228. So, the area between these z-scores is: \[ 0.9772 - 0.0228 = 0.9544 \] Now, multiply by the total number of students: \[ 0.9544 \times 1400 \approx 1336.16 \] Rounding to the nearest person, about **1336 students** have an IQ score between 90 and 110. For part c, to find how many have an IQ score over 115, we can calculate the z-score for 115: \[ z = \frac{(115 - 100)}{5} = 3 \] The area below \( z = 3 \) is approximately 0.9987. Thus, the area above it (for scores over 115) is: \[ 1 - 0.9987 = 0.0013 \] Finally, multiplying by the total number of students gives: \[ 0.0013 \times 1400 \approx 1.82 \] Therefore, rounding to the nearest person, about **2 students** have an IQ score over 115.