Quick questions on Special relativity time dilation: WACE Year 12 Physics Unit 4

A clock measures the shortest time, the proper time $t_0$, in the frame where it is at rest. An observer moving relative to that clock measures a longer time

An object has its greatest length, the proper length $L_0$, in its own rest frame. An observer past whom it moves measures a contracted length along the direction of motion

Muons created high in the atmosphere have a short half-life and should mostly decay before reaching the ground at their speed. Yet far more arrive than classical physics predicts. From the ground frame, the muons' internal clocks run slow (time dilation), so they live longer. From the muon frame, the atmosphere is thin (length contraction), so there is less distance to cross.

The proper time is measured by a clock present at both events (the moving object's clock); the proper length is measured in the object's rest frame. Time dilation multiplies the proper time by $\gamma$; length contraction divides the proper length by $\gamma$. Choosing the wrong frame for the proper value inverts the result.