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.
