To find the missing sample mean, we can use the formula for the average of the sample means. Let's denote the missing sample mean as \( x \).
The estimated population mean is calculated as follows:
\[ \text{Population Mean} = \frac{\text{Sum of all sample means}}{\text{Number of samples}} \]
In this case, we know:
- The estimated population mean is \( 4.2 \).
- There are 6 samples in total (including the missing one).
Thus, we have:
\[ 4.2 = \frac{4.8 + 3.2 + 4.6 + 5.3 + 3.9 + x}{6} \]
First, we can calculate the sum of the known sample means:
\[ 4.8 + 3.2 + 4.6 + 5.3 + 3.9 = 21.8 \]
Now we can substitute this into the equation:
\[ 4.2 = \frac{21.8 + x}{6} \]
Next, we multiply both sides by 6 to eliminate the fraction:
\[ 25.2 = 21.8 + x \]
Now, subtract \( 21.8 \) from both sides to solve for \( x \):
\[ x = 25.2 - 21.8 = 3.4 \]
Thus, the missing sample mean is \( \boxed{3.4} \).