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------------------------------------------------ Find the mean, [tex]X[/tex], of the data set: 3, 4, 5, 7, 10, 12, 15.

[tex]X = \, ?[/tex]

To find the missing element [tex]\( x \)[/tex] in the data set that makes the mean equal to 13, we need to follow these steps:

1. Understand the Problem: We have a data set: 3, 4, 5, 7, 10, 12, 15, and an unknown element [tex]\( x \)[/tex]. We need to adjust the data set to have a mean of 13.

2. Calculate the Current Sum: First, calculate the sum of the given numbers:
[tex]\[
3 + 4 + 5 + 7 + 10 + 12 + 15 = 56
\][/tex]

3. Determine the Number of Elements: Currently, there are 7 elements in the data set. Including the unknown element [tex]\( x \)[/tex], the total number of elements will be 8.

4. Set Up the Mean Equation: For the mean to be 13, the total sum of all 8 elements divided by the number of elements (8) should equal 13. Therefore:
[tex]\[
\frac{56 + x}{8} = 13
\][/tex]

5. Solve for [tex]\( x \)[/tex]:
- Multiply both sides by 8 to eliminate the fraction:
[tex]\[
56 + x = 104
\][/tex]
- Subtract 56 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = 104 - 56 = 48
\][/tex]

Therefore, the missing element [tex]\( x \)[/tex] in the data set that makes the mean equal to 13 is 48.