Are we sure we're referring to the same 'dead space'? The question said that anatomical dead space does not participate in gas exchange. This is true. Deadspace represents areas which are ventilated but not perfused.
Is that the same thing as the air that does not get expelled during respiration? There is always gas in the deadspace regardless of the point of ventilation. This has to be true, or else the respiratory tract would collapse after every breath. The compositions of the air change, however. When you breathe in, the deadspace is left filled with atmospheric air that did not make it into the alveoli. This air would have atmospheric composition. When you breathe out, however, the air which fills the deadspace at end expiration is the air which came from the alveoli. Thus it has alveolar gas concentrations. These would be lower pO2 and higher pCO2.
I was thinking of the dead space referred to in the question as parts of the respiratory tract, such as the trachea, bronchi, etcetera that do not interface with the capillaries and participate in gas exchange. This is correct.
Therefore, the partial pressures of this air is identical to the atmospheric partial pressures, which is higher in CO2 than O2 than the alveolar air, right? Atmospheric air has a lower pCO2 than alveolar air. If this weren't true, CO2 would go into the blood instead of coming out. Alveolar air is lower in O2 than atmospheric air because some of it has diffused into the blood