pling Distributions 1. Forest type 1 has an average diameter of 35 cm and a standard deviation of 6 cm. Forest type 2 has an average diameter of 25 cm and a standard deviation of 5 cm. The distribution of the diameters is normal in both types. a. What is the probability that a sample mean will be greater than 36 cm from type 1 if 20 trees are measured? b. What is the probability that a sample variance is less than 19 cm2 from type 1 if 20 trees are measured? c. Find the probability that the sample mean computed from 20 measurements from type 1 will exceed the sample mean computed from 25 measurements from type 2 by at least 11 cm. d. Find the probability that the ratio of sample variances (S1 2/S2 2) computed from 10 measurements from type 1 (S1 2) and 8 measurements from type 2 (S2 2) will exceed 5.4. 2. The probability of finding a bark beetle infested tree in forest type 1is 0.30 and in forest type 2 is 0.35. a. If a sample of 50 trees is taken from forest type 1, what is the probability that the proportion of beetle attack is less than 0.20? b. If a sample of 35 trees is taken from forest type 1and another sample of 40 from forest type 2, what is the probability that the proportion of beetle attacked trees from type 2 will exceed the proportion from type 1 by at least 0.1? 3. The following are some of the data collected in one of the mensuration exercises. The measurements are volumes in m3/ha for 24 different plots: Column 1 Column 2 736 1009 1560 1413 990 1528 924 1079 1042 1207 1360 1770 936 826 1618 1573 1221 1259 1033 1564 1288 896 555 1462 a. Assume that each column of data comes from a different forest type. Is it very likely to get a variance ratio S1 2/S2 2 as big as, or greater than what you got if you assume that σ1 2 = σ2 2 . Can you assume that σ1 2 = σ2 2 with a good probability? b. You don't know the population variance of the two forest types, but you are told that µ1 = µ2 . What is the probability that, if you take 12 plots from each type: 6/24/2015 Assignment Instructions – 2015SA-FRST231-... data:text/html;charset=utf-8,%3Cdiv%20id%3D%22container%22%20class%3D%22clearfix%20full_width%22%20style%3D%22margin%3A%200px%3B%20pad… 2/3 i. x̄2 - x̄1 ≤ 200? ii. 100 ≤ x̄2 - x̄1 ≤ 200? iii. to get a difference as big as you have or greater? Estimation of Parameters 1. Measurements (in cm) of diameter at breast height (dbh) were taken of 15 trees from a forest type: 17.2 18.9 16.2 25.8 24.5 25.1 27.8 29.2 16.1 18.4 25.7 23.2 30.1 17.2 27.9 a. Find the 90%, 95% and 99% confidence interval for the mean of all diameters of this stand. Draw your conclusions. b. Find the number of observations you have to take to get a 1 cm sampling error if you want 95% confidence in your estimation. c. Find the 95% confidence limits for the variance of all diameters of this stand. 2. a. A regeneration survey has indicated that 75 of the 90 plots examined were stocked. Find the 99% confidence limits for the fraction of the total area being stocked. b. How large a sample is needed if we wish to be 95% confident that our sample proportion will be within 0.05 of the true fraction? 3. Another survey from another area indicated 75% stocking from 100 plots. Is this different from the area described in Question 5? Use 90% and 95% confidence limits. 4. The volume in m3/ha in a mature Douglas-fir stand was estimated from the same 10 plots independently by two crews (thus giving "paired observations"). Plot Crew #1 Crew #2 1 875 910 2 959 878 3 475 480 4 589 495 5 925 1021 6 1200 980 7 971 1002 8 421 410 9 892 850 10 728 620 Are the two estimates different? Use 95% confidence. 5. A study was carried out to compare the salaries of female and male first year forestry students during the summer of 1980. The random samples taken from the two populations are summarized below: Salary/Month Female Male x̄ $1,125 $1,310 s $300 $130 6/24/2015 Assignment Instructions – 2015SA-FRST231-... data:text/html;charset=utf-8,%3Cdiv%20id%3D%22container%22%20class%3D%22clearfix%20full_width%22%20style%3D%22margin%3A%200px%3B%20pad… 3/3 n 10 21 Compute the 90% and 99% confidence intervals for the difference between the two population means. Draw your conclusions.