Step 1: Identify the formula

The arithmetic mean of a set of observations is the sum of all observations divided by the total number of observations.

$\text{Arithmetic Mean} = \frac{\text{Sum of Observations}}{\text{Total Number of Observations}}$

Step 2: Calculate the sum of observations

Add all the given numbers together: $39 + 49 + 59 + 60 + 70 + 80 + 90 + 93 + 90$.

$\sum x_i = 710$

Step 3: Count the number of observations

Counting the individual observations provided, we find there are $9$ values in total ($n = 9$).

$n = 9$

Step 4: Compute the final arithmetic mean

Divide the sum by the number of observations: $710 / 9 \approx 78.88$, which rounds to $79$ (matching the integer values of the options).

$\text{Mean} = \frac{710}{9} = 78.88... \approx 79$