It is claimed that two cesium clocks, if allowed to run for $100$ years, free from any disturbance, may differ by only about $0.02s$ . What does this imply for the accuracy of the standard cesium clock in measuring a time-interval of $1\,s$ ?

Hint: Here we have to first find the error for $100$ years, then the error for $1\,s$ .
This will give us the accuracy of the standard cesium clock.
Cesium clock is the most reliable type of clock yet to be produced. This system takes use of the transitions between the spin states of the cesium nucleus and generates a rhythm that is so normal that it has been used to decide the time.

Complete step by step answer:
Inside the Cesium Atomic clock, the Cesium atoms are encircled down a tunnel, where they travel by radio waves. If this frequency is only $9192631770$ cycles per second, the cesium atoms "resonate" and change their energy status. The detector feeds information back to the radio wave generator.
The theory of the operation of the atomic clock is based on atomic physics; it calculates the electric signal produced by electrons in atoms as they change their energy levels. Early atomic clocks were based on masers at room temperature. Since $2004$ ,more precise atomic clocks have first cooled the atoms to near absolute zero temperature by cooling them down with lasers and probing them through atomic fountains in a microwave-filled cavity.
Given,
Error in $100$ years $ = 0.02s$
Error in $1\,s$
$
 = \dfrac{{0.02}}
{{100 \times 365 \times \dfrac{1}
{4} \times 24 \times 60 \times 60}} \\
 = 7.9 \times {10^{ - 13}} \simeq {10^{ - 12}} \\
$

Hence, the accuracy of the standard cesium clock in measuring a time-interval of $1\,s$ is ${10^{ - 12}}$ .

Note:
Here we have to be careful that the number of years is $100$ . So, for measuring $1\,s$ we have to multiply numerous numbers to make it equivalent to one second. By calculating the oscillation of the atoms, the atomic clocks remain reliable, but they are not flawless. They suffer an error of one second per one-hundred million years or so.