4-1PROBLEM 4.1 KNOWN:N > 1000; x= 9.2 units ; Sx= 1.1 units FIND:Range of x in which 50% of all measurements should fall. ASSUMPTIONS:Measurand follows a normal density function Data set sufficiently large such that x≈x’ and Sx≈σSOLUTION By assuming that the data is sufficiently large such that its population behaves as an infinite population, we can find the interval defined by x' - z1σ≤xi≤x' + z1σas follows. We can find P(x' + z1σ) from the one-sided integral solution to This solution is given in Table 4.3 for p(z1) = 0.25 (one half of the 50% probability sought) as z1= 0.674. Then, we dβe][21)P(z12Z0/2β1/21∫−π=
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4-2should expect that 50% of the xi values lie in the interval given by 9.2 – 0.7425 ≤xi≤9.2 + 0.7425 (50%) COMMENT We can see from Table 4.4 that as N becomes large the value for t approaches a value given by z1.