To find the missing sample mean, we can use the formula for the estimated population mean, which is the average of the sample means.
Let the missing sample mean be \( x \). The formula for the population mean is:
\[ \text{Population Mean} = \frac{\text{Sum of sample means}}{\text{Number of samples}} \]
We know the estimated population mean is 14.1 and there are four samples. Therefore, we can write:
\[ 14.1 = \frac{14.8 + 13.2 + 14.6 + x}{4} \]
Now, let's calculate the sum of the known sample means:
\[ 14.8 + 13.2 + 14.6 = 42.6 \]
Now, substitute this into the equation:
\[ 14.1 = \frac{42.6 + x}{4} \]
To eliminate the fraction, multiply both sides by 4:
\[ 4 \times 14.1 = 42.6 + x \]
Calculating \( 4 \times 14.1 \):
\[ 56.4 = 42.6 + x \]
Now, isolate \( x \) by subtracting 42.6 from both sides:
\[ x = 56.4 - 42.6 \] \[ x = 13.8 \]
Thus, the missing sample mean is \( \boxed{13.8} \).