I believe that what you are confused on is the symmetry of time dilation and length contraction. Meaning that if you observe an object to be moving relative to you, it is also appropriate to consider that you are moving relative to it, meaning that special relativity applies "both" ways. Thus, its critical to define what reference frame is being considered.
First lets define who measures the proper length and time:
Proper length is the length measured by the observer for whom the endpoints of the length remain fixed in time and space (in the referred to reference frame)
Proper time is the time interval measured by the observer who sees the two events happen at the same position in space (in the referred to reference frame)
The question refers to the reference frame of the scientist, consequently in this reference frame he is motionless on the ground, and the muon is moving relative to him, and from this we can conclude two things:
Since the scientist is motionless in such a reference frame and the muon is moving relative to him, he is the observer with the "privilege" of measuring the endpoints of the altitude to be fixed, since no matter what time of the event you take the endpoints will remain in the same position relative to him (unlike the muon). Thus, he measures the proper length.
Moreover, since we are referring to the scientists reference frame we know that the muon is moving relative to him, thus muon's position relative to him is a subject of time, however for the muon in such a reference frame we know that from its "perspective" it sees the events as happening in the same position relative to it, thus it measures the proper time, with respect to the scientists reference frame
In conclusion an object which is in motion relative to another object in a certain reference frame always measures the proper time, as well as the dilated length, in that reference frame.