determining a missing value\nwhich value should be added to this data set to produce a mean of 13?\n10 11 12…

Explanation:

Step1: Calculate sum of original data set

The original data - set is (10,11,12,13,14,15,16,17,18,19,20). The sum of an arithmetic series (S_n=\sum_{i = 1}^{n}a_i), and here (n = 11), (a_i) is the (i) - th number in the set. Using the formula for the sum of an arithmetic series (S=\frac{n(a_1 + a_n)}{2}), where (a_1 = 10), (a_n=20), (n = 11), so (S=\frac{11\times(10 + 20)}{2}=11\times15 = 165).

Step2: Let the missing value be (x)

We want the new mean of the data - set with (n'=12) (the original 11 values plus the new value (x)) to be 13. The formula for the mean (\bar{y}=\frac{\text{Sum of all values}}{\text{Number of values}}). So (13=\frac{165 + x}{12}).

Step3: Solve for (x)

Multiply both sides of the equation (13=\frac{165 + x}{12}) by 12: (13\times12=165 + x). Then (156=165 + x). Subtract 165 from both sides: (x=156 - 165=-9).

Answer:

(-9)