To find the missing sample mean, we can use the formula for the estimated population mean based on the sample means. The estimated population mean is the average of all the sample means.
Let \( x \) be the missing sample mean. We know that:
\[ \text{Estimated Population Mean} = \frac{\text{Sum of Sample Means}}{\text{Number of Samples}} \]
In this case, the estimated population mean is 14.1, and there are four sample means, three of which are provided (14.8, 13.2, and 14.6). Therefore, we can set up the equation:
\[ 14.1 = \frac{14.8 + 13.2 + 14.6 + x}{4} \]
Now, first, we will calculate the sum of the known sample means:
\[ 14.8 + 13.2 + 14.6 = 42.6 \]
Substituting this into the equation gives:
\[ 14.1 = \frac{42.6 + x}{4} \]
Now multiply both sides by 4 to eliminate the fraction:
\[ 4 \times 14.1 = 42.6 + x \]
Calculating \( 4 \times 14.1 \):
\[ 56.4 = 42.6 + x \]
Now, isolate \( x \) by subtracting 42.6 from both sides:
\[ x = 56.4 - 42.6 \]
Calculating \( 56.4 - 42.6 \):
\[ x = 13.8 \]
Thus, the missing sample mean is \( \boxed{13.8} \).