Final answer:
The absolute difference between the two remaining observations is calculated by first finding their sum using the given mean and then considering pairs that add up to this sum while conforming to the given variance. Option D is correct.
Step-by-step explanation:
The student is asking for the absolute difference between the two remaining observations in a dataset where the mean is 10 and the variance is 13.5, and the values of six of the observations are given.
To solve this, we can use the fact that the sum of all observations is 8 times the mean (since there are 8 observations in total). So the total sum of all observations is 8 * 10 = 80. The sum of the six given observations (5 + 7 + 10 + 12 + 14 + 15) is 63. Therefore, the sum of the remaining two observations is 80 - 62 = 18.
To find their absolute difference, consider the possible pairs of numbers that sum up to 18, and then use the given variance to filter out the correct pair. Variance is the average of the squared differences from the mean. Since the calculated variance is 13.5, the squared differences of the remaining two observations from the mean must sum up in such a way that when added to the squared differences of the known observations and divided by 8, results in 13.5.
After performing the variance calculations with different pairs, you'll find that the correct pair of remaining observations which confirms the given variance is 9 and 8.