To find the missing sample mean, we can use the relationship between the sample means and the estimated population mean.
First, let's sum the known sample means:
\[ 9.7 + 7.8 + 8.5 + 9.3 + 9.2 + 9.1 + 8.7 + 8.6 + 7.7 \]
Calculating this gives:
\[ 9.7 + 7.8 = 17.5\ 17.5 + 8.5 = 26.0\ 26.0 + 9.3 = 35.3\ 35.3 + 9.2 = 44.5\ 44.5 + 9.1 = 53.6\ 53.6 + 8.7 = 62.3\ 62.3 + 8.6 = 70.9\ 70.9 + 7.7 = 78.6\ \]
So, the total of the known sample means is 78.6.
We know that the average of the 10 sample means is equal to the estimated population mean, which is 8.7. Thus, we can find the total sum of all 10 sample means:
\[ \text{Average} = \frac{\text{Sum of sample means}}{\text{Number of samples}} \implies \text{Sum of sample means} = \text{Average} \times \text{Number of samples} \]
\[ \text{Sum of sample means} = 8.7 \times 10 = 87 \]
Next, let \( x \) be the missing sample mean. Therefore, we can write the equation:
\[ 78.6 + x = 87 \]
Solving for \( x \):
\[ x = 87 - 78.6 = 8.4 \]
Thus, the missing sample mean is 8.4.