Sure, let's determine the new mean when 17 is added to the given data set.
1. Original Data Set and Mean: The original data set is: [tex]\[
12, 30, 19, 27, 21, 35
\][/tex] The mean of this data set is given as 24.
2. Calculate the Original Sum: The mean ([tex]\( \overline{x} \)[/tex]) of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers. [tex]\[
\overline{x} = \frac{\sum x_i}{n}
\][/tex] Here, [tex]\( \overline{x} = 24 \)[/tex] and [tex]\( n = 6 \)[/tex] (since there are 6 numbers in the original data set).
To find the original sum ([tex]\( \sum x_i \)[/tex]): [tex]\[
24 = \frac{\sum x_i}{6}
\][/tex] Multiplying both sides by 6 gives: [tex]\[
\sum x_i = 144
\][/tex]
3. Add the New Value to the Sum: Now, we add the new value (17) to the original sum: [tex]\[
\text{New Sum} = 144 + 17 = 161
\][/tex]
4. Determine the New Count of Numbers: Initially, there were 6 numbers. By adding one more number, the new count becomes: [tex]\[
n_{\text{new}} = 6 + 1 = 7
\][/tex]
5. Calculate the New Mean: Using the new sum and the new count of numbers, the new mean ([tex]\( \overline{x}_{\text{new}} \)[/tex]) is: [tex]\[
\overline{x}_{\text{new}} = \frac{161}{7}
\][/tex]
6. Simplify the New Mean: Performing the division: [tex]\[
\overline{x}_{\text{new}} = 23.0
\][/tex]
So, the new mean after adding 17 to the data set is 23.0.
