To solve the problem, we need to find the value of \( x \) such that the median of the given observations is 24, and then calculate the mean of the data.
### Step 1: Identify the given observations
The observations are:
\[ 10, 11, 13, 17, x + 2, x + 4, 31, 33, 36, 42 \]
### Step 2: Determine the number of observations
There are 10 observations in total.
### Step 3: Calculate the median for an even number of observations
For an even number of observations, the median is the average of the \( \frac{n}{2} \)th term and the \( \left(\frac{n}{2} + 1\right) \)th term. Here, \( n = 10 \), so we need to find the 5th and 6th terms.
### Step 4: Arrange the observations in ascending order
To find the median, we need to arrange the observations. The terms \( x + 2 \) and \( x + 4 \) will depend on the value of \( x \).
### Step 5: Set up the equation for the median
The 5th term is \( x + 2 \) and the 6th term is \( x + 4 \). The median is given as 24, so we have:
\[
\frac{(x + 2) + (x + 4)}{2} = 24
\]
### Step 6: Solve for \( x \)
Simplifying the equation:
\[
\frac{2x + 6}{2} = 24
\]
\[
2x + 6 = 48
\]
\[
2x = 42
\]
\[
x = 21
\]
### Step 7: Substitute \( x \) back into the observations
Now substitute \( x = 21 \) into the observations:
- \( x + 2 = 21 + 2 = 23 \)
- \( x + 4 = 21 + 4 = 25 \)
The observations now are:
\[ 10, 11, 13, 17, 23, 25, 31, 33, 36, 42 \]
### Step 8: Calculate the mean
To find the mean, we sum all the observations and divide by the number of observations (10):
\[
\text{Sum} = 10 + 11 + 13 + 17 + 23 + 25 + 31 + 33 + 36 + 42
\]
Calculating the sum:
\[
10 + 11 = 21
\]
\[
21 + 13 = 34
\]
\[
34 + 17 = 51
\]
\[
51 + 23 = 74
\]
\[
74 + 25 = 99
\]
\[
99 + 31 = 130
\]
\[
130 + 33 = 163
\]
\[
163 + 36 = 199
\]
\[
199 + 42 = 241
\]
Now, divide the sum by the number of observations:
\[
\text{Mean} = \frac{241}{10} = 24.1
\]
### Final Answer
The mean of the data is \( 24.1 \).
