The distance between two points needs to be measured in mete

The distance between two points needs to be measured, in meters. The true distance is 25 meters, but due to measurement error we can\'t measure the distance exactly. Instead, we will observe a value of 25 + epsilon, where the error iris distributed N(0,0.16). Find the probability that the observed distance is within 0.8 meters of the true distance (25 meters). Give your answer in terms of (the CDF of Normal (0,1)).

Solution

Observed distance is between ( 25 - 0.8) = 24.2 and ( 25+ 0.8) = 25.8...

P [ observed distance is within 0.8 metres of true distance ]
= P [ observed distance <= 25.8 ] - P [ observed distance <= 24.2 ]
= P [ Z <= ( 25.8 - 25) / sqrt ( 0.16) ] - P [ Z <= ( 24.2 - 25) / sqrt ( 0.16) ]
= P [ Z <= 2 ] - P [ Z <= - 2 ] = 2* P [ Z <=2 ] - 1 .......(ans)
where Z = cdf of Normal( 0,1)...

=  9545 ( if you want this too,just in case)