Final answer:
Replacing a value in the dataset with a larger one farther from the mean will increase the standard deviation. The correct answer is that the standard deviation will increase because the value 12 increases variation from the mean, which causes an increase in standard deviation.
Step-by-step explanation:
The question is asking how replacing a value in a data set with a larger value affects the standard deviation. Standard deviation is a measure of the amount of variation or dispersion of a set of values. When a number in a data set is replaced by a value that is further away from the mean of the set, like replacing 7 with 12 in this case, the variance increases because the new number (12) increases the disparity from the mean. Since standard deviation is the square root of variance, an increase in variance results in an increase in standard deviation.
Initially, the mean of the set {1, 4, 7, 2, 6} is 4. Replacing 7 with 12 would result in the new set {1, 4, 12, 2, 6}, which has a higher mean. As the value 12 is farther from the new mean than 7 was from the original mean, the deviation of the values from the mean has increased, thereby increasing the standard deviation.
Therefore, the correct answer is: c. The standard deviation will increase because 12 is a larger value.