Final answer:
The calculation of average sick days from the sample data is 8.375, which is less than the believed average of 10 sick days per employee. The personnel department's skepticism is supported by this sample, but further statistical testing would be required for a definitive conclusion.
Step-by-step explanation:
The question asks whether the personnel department should believe that the mean number of sick days taken by employees is 10, given sample data. To determine this, we calculate the mean of the provided data points. Adding the numbers of sick days together (12 + 4 + 15 + 3 + 11 + 8 + 6 + 8), we get a total of 67 sick days. Dividing this total by the number of employees surveyed (8), we get an average of 8.375 sick days per employee. Based on this sample mean, it is less than the believed average of 10 days, suggesting that the personnel department might be justified in questioning the original figure. However, to make a conclusive decision, a statistical hypothesis test, such as a t-test, could be performed to determine if the difference is statistically significant.