Final answer:
a) The weight representing the 35th percentile is 1096 pounds. b) The weight representing the 91st percentile is 1204 pounds. c) The IQR of the weights of these steers is 72 pounds.
Step-by-step explanation:
a) To find the weight that represents the 35th percentile, we need to find the z-score that corresponds to a cumulative probability of 0.35. Using the z-score formula z = (x - mean) / standard deviation, we can solve for x: (x - 1120) / 63 = -0.385. Rearranging the equation, we get x - 1120 = -0.385 * 63 = -24.255. Solving for x, we get x = 1120 - 24.255 = 1095.745. Rounding to the nearest whole number, the weight representing the 35th percentile is 1096 pounds.
b) Similarly, to find the weight that represents the 91st percentile, we need to find the z-score that corresponds to a cumulative probability of 0.91. Using the z-score formula, we can solve for x: (x - 1120) / 63 = 1.34. Rearranging the equation, we get x - 1120 = 1.34 * 63 = 84.42. Solving for x, we get x = 1120 + 84.42 = 1204.42. Rounding to the nearest whole number, the weight representing the 91st percentile is 1204 pounds.
c) The interquartile range (IQR) is the difference between the third quartile and the first quartile. Since the first quartile represents the 25th percentile and the third quartile represents the 75th percentile, we can use the same approach as above to find their corresponding weights. The weight representing the 25th percentile is 1084 pounds and the weight representing the 75th percentile is 1156 pounds. Therefore, the IQR is 1156 - 1084 = 72 pounds.