To find the missing sample mean, we can use the formula for the estimated population mean, which is the average of all the sample means.
Given:
- Estimated population mean (M) = 14.1
- Sample means = 14.8, 13.2, 14.6, and one missing sample mean (let's call it \( x \))
The formula for the mean of the four sample means is:
\[ M = \frac{14.8 + 13.2 + 14.6 + x}{4} \]
Setting this equal to the estimated population mean gives us:
\[ 14.1 = \frac{14.8 + 13.2 + 14.6 + x}{4} \]
Now let's calculate the sum of the known sample means:
\[ 14.8 + 13.2 + 14.6 = 42.6 \]
Now we can 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 the left side:
\[ 56.4 = 42.6 + x \]
Now, solve for \( x \):
\[ x = 56.4 - 42.6 \] \[ x = 13.8 \]
Thus, the missing sample mean is 13.8.