Step 1 :Given the data set of the amounts of time (in months) that the employees of a restaurant have been working at the restaurant: \[\{1, 5, 6, 8, 11, 14, 17, 46, 61, 90, 99, 126, 143, 167\}\]
Step 2 :The mean of a set of numbers is calculated by adding all numbers in the set and then dividing by the count of numbers in the set.
Step 3 :First, add all the months that the employees have been working at the restaurant: \[1 + 5 + 6 + 8 + 11 + 14 + 17 + 46 + 61 + 90 + 99 + 126 + 143 + 167 = 794\]
Step 4 :Then, divide the total by the number of employees, which is 14: \[\frac{794}{14} = 56.714285714285715\]
Step 5 :Round the result to one more decimal place than the original data values, which gives approximately 56.7.
Step 6 :Final Answer: The mean for the given sample data is approximately \(\boxed{56.7}\) months.