36. The normal distribution is commonly used to model the variability expected w

Question

36. The normal distribution is commonly used to model the variability expected when making measurements (Taylor, J. R, An Introduction to Error Analysis: The Study of Uncertainties in Physicai Measurements, University Science Books, Sausalito, CA, 1997). In this context, a measured quantity x is assumed to have a normal distribution whose mean is assumed to be the “true" value of the object being measured uring instrument deter- The precision of the meas mines the standard deviation of the distribution. a. If the measurements of the length of an object have a normal probability distribution with a standard deviation of 1 mm, what is the prob- ability that a single measurement will lie within 2 mm of the true length of the object?

Explanation / Answer

as z score =(X-mean)/std deviation

a)here probability that a single measurement will lie with in 2 mm of true length =P(-2/1<Z<2/1) =P(-2<Z<2)

=0.97725-0.02275 =0.9545

b) probability that a single measurement will lie with in 2 mm of true length

=P(-2/0.5<Z<2/0.5)=P(-4<Z<4)=0.999968-0.00000=0.999937

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