We often say that the probability of a measurement in the ra

We often say that the probability of a measurement in the range X ± 2? is about 95%. What is this probability to four significant figures?

What range X ± t? actually has probability equal to 95.00%? (Give t to three significant figures. The needed probabilities and formulas can be found below.)

If the measurement of a continuous variable x is subject to many small errors, all of them random, the expected distribution of results is given by the normal, or Gauss, distribution, where X is the true value of x, and is the standard deviation. The integral of the normal distribution function, Gx,o(r)dx, is called the nor- mal error integral, and is the probability that a measurement falls between x=a and x = b, Table A shows this integral for a =X-to and b =X+ to. This gives the probabil- ity of a measurement within t standard deviations on either side of X, -22 dz. X-to 2T This function is sometimes denoted erf(t), but this notation is also used for a slightly different function. The probability of a measurement outside the same interval can be found by subtraction; Prob(outside to) 100% -Prob(withinto.

Solution

1. 68.27% . Probability is 0.6827. It the value in the row 1.0 and column 00.

2. 1.96. From teh table we find teh value 95% in the row 1.9 and column 06