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University of California Los Angeles Probability and Statistical Inference Worksheet

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7.5-1. Let Y1 < Y2 < Y3 < Y4 < Y5 < Y6 be the order statistics of a random sample of size n = 6 from a distri-bution of the continuous type having (100p)th percentile πp. Compute

(a) P(Y2 <π0.5 < Y5). (b) P(Y1 <π0.25 < Y4). (c) P(Y4 <π0.9 < Y6).

7.5-4. Let m denote the median weight of “80-pound” bags of water softener pellets. Use the following random sample of n = 14 weights to find an approximate 95% confidence interval for m:

80.51 80.28 80.40 80.35 80.38 80.28 80.27 80.16 80.59 80.56 80.32 80.27 80.53 80.32

(a) Find a 94.26% confidence interval for m.

(b) The interval (y6, y12) could serve as a confidence inter-val for π0.6. What is its confidence coefficient?

7.5-5. A biologist who studies spiders selected a random sample of 20 male green lynx spiders (a spider that does not weave a web, but chases and leaps on its prey) and measured the lengths (in millimeters) of one of the front legs of the 20 spiders. Use the following measurements toconstruct a confidence interval for m that has a confidence coefficient about equal to 0.95:

15.10 16.40 13.55 15.75

13.60 16.45 14.05 17.05 15.25

17.75 15.40 16.80

20.00 15.45 16.95 19.05 16.65 16.25 17.55 19.05

7.5-12. Let Y1 < Y2 < ··· < Y8 be the order statistics of eight independent observations from a continuous-type distribution with 70th percentile π0.7 = 27.3.(a) Determine P(Y7 < 27.3). (b) Find P(Y5 < 27.3 < Y8).

8.1-2. Assume that the weight of cereal in a “12.6-ounce box” is N(μ, 0.22). The Food and Drug Association (FDA) allows only a small percentage of boxes to contain less than 12.6 ounces. We shall test the null hypothesis H0: μ = 13 against the alternative hypothesis H1: μ< 13. (a) Use a random sample of n = 25 to define the test statistic and the critical region that has a significance level of α = 0.025.

(b) If x = 12.9, what is your conclusion? (c) What is the p-value of this test?

8.1-3. Let X equal the Brinell hardness measurement of ductile iron subcritically annealed. Assume that the distri-bution of X is N(μ, 100). We shall test the null hypothesis H0: μ = 170 against the alternative hypothesis H1: μ> 170, using n = 25 observations of X.

(a) Define the test statistic and a critical region that has a significance level of α = 0.05. Sketch a figure showing this critical region.

(b) A random sample of n = 25 observations ofX yielded the following measurements:

170 167 174 179 179 156 163 156 187 156 183 179 174 179 170 156 187 179 183 174 187 167 159 170 179

Calculate the value of the test statistic and state your conclusion clearly.

(c) Give the approximate p-value of this test.

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