(a) One-sided tolerance intervals for the distribution of a ran – dom variable X are reasonably accessible because, in fact, they're just confidence intervals for the corresponding percentile. That's not true for TWo-sided intervals which require a more subtle analysis. Show that the 90 % tolerance interval for 93 % of the values of “fill” discussed in the text above is just a one-sided 90 % confidence interval for the 93th percentile of the distribution of “fill.” Do this in terms of a picture for the distribution of X = “fill.”(b) The endpoint of a one-sided tolerance interval to encompass a proportion p of the values of any random variable X will be determined by an order statistic, X(k), from a sample We'll assume X has a negligible chance to take on any of the percentiles under discussion and that np, nq ≥ 5.Show that k is determined bywhere the value of Z is determined by P(Z α.(c) In the level II answer for part (b), why is the picture not a normal curve? And if it's not a normal curve, how do we end up with a value of Z in the formula? (d) Show that the formula in part (b) gives the “right answer” if p = . (e) Show that the lower and upper tolerance limits taken together do NOT give the endpoints of a two-sided tolerance interval.
(a) One-sided tolerance intervals for the distribution of a ran – dom variable X are reasonably…
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