Final answer:
To find the value of the largest integer in data set C, we need to determine the sum of the remaining integers and ensure it is the least multiple of 12 that is greater than the sum of the given 11 integers. To do this, we can divide the sum of the given 11 integers by 12 and round up to the nearest whole number. This will give us the minimum sum of the remaining integers, and by adding it to the sum of the given 11 integers, we can find the minimum sum of the entire data set.
Step-by-step explanation:
To find the value of the largest integer in data set C, we need to find the mean of the entire data set. We know that the mean of the given 11 integers is 54. If the mean of the entire data set is an integer greater than 54, it means that the sum of the remaining integers in data set C must be larger than the sum of the given 11 integers.
Since the mean is an integer, the total sum of the data set must be a multiple of 12 (the number of integers in the data set) to have an integer mean. To find the largest integer in data set C, we need to determine the sum of the remaining integers and ensure it is the least multiple of 12 that is greater than the sum of the given 11 integers.
Let's calculate the sum of the given 11 integers: 54+57+56+55+49+50+52+58+51+59+53 = 594. To find the least multiple of 12 that is greater than 594, we can divide 594 by 12 and round up to the nearest whole number: 594 ÷ 12 = 49.5. Rounding up gives us 50. Therefore, the sum of the remaining integers in data set C must be at least 50.
To find the largest integer in data set C, we can now subtract the sum of the given 11 integers from the sum of the entire data set. The sum of the entire data set (including the largest integer) is at least 594 (the sum of the given 11 integers) plus 50 (the minimum sum of the remaining integers). Therefore, the largest integer in data set C is at least 644.