6.3-1. Some biology students were interested in analyz-ing the amount of time that bees spend gathering nectar in flower patches. Thirty-nine bees visited a high-density flower patch and spent the following times (in seconds) gathering nectar:

235 210 95 146 195 840 185 610 680 990 146 404 119 47 9 4 10 169 270 95 329 151 211 127 154 35 225 140 158 116 46 113 149 420 120 45 10 18 105

(a) Find the order statistics. (b) Find the median and 80th percentile of the sample.

(c) Determine the first and third quartiles (i.e., 25th and 75th percentiles) of the sample.

6.3-7. Let Y1 < Y2 < ··· < Y19 be the order statistics of n = 19 independent observations from the exponential distribution with mean θ.

(a) What is the pdf of Y1?

(b) Using integration, find the value of E[F(Y1)], where F is the cdf of the exponential distribution.

6.3-13. Some measurements (in mm) were made on spec-imens of the spider Sosippus floridanus, which is native to Florida. Here are the lengths of nine female spiders and nine male spiders.Female spiders 11.06 13.87 12.93 15.08 17.82 14.14 12.26 17.82 20.17

Male spiders

12.26 11.66 12.53 13.00 11.79 12.46 10.65 10.39 12.26

(a) Construct a q-q plot of the female spider lengths. Do they appear to be normally distributed?

(b) Construct a q-q plot of the male spider lengths. Do they appear to be normally distributed?

6.3-14. An interior automotive supplier places several electrical wires in a harness. A pull test measures the force required to pull spliced wires apart. A customer requires that each wire that is spliced into the harness withstand a pull force of 20 pounds. LetXequal the pull force required to pull a spliced wire apart. The following data give the values of a random sample of n = 20 observations of X:

28.8 24.4 30.1 25.6 26.4 23.9 22.1 22.5 27.6 28.1 20.8 27.7 24.4 25.1 24.6 26.3 28.2 22.2 26.3 24.4

(a) Construct a q-q plot, using the ordered array and the corresponding quantiles of N(0, 1).

(b) Does it look like X has a normal distribution?

6.4-1. Let X1,X2,…,Xn be a random sample from N(μ, σ2), where the mean θ = μ is such that −∞ < θ< ∞ and σ2 is a known positive number. Show that the maximum likelihood estimator for θ is θ = X.

6.4-2. A random sample X1,X2,…,Xn of size n is taken fromN(μ, σ2), where the variance θ = σ2 is such that 0 < θ< ∞and μ is a known real number. Show that the maxi-mum likelihood estimator for θ isθ = (1/n)

n i=1 (Xi−

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