Quiz 8 (retest)

Time:  20 minutes.  This quiz is open book, open notes.

1.    Samples of sediment in a river basin yielded values of 8, 11, 36, and 228 mg/Kg (ppm) arsenic.  To analyze the results, you will model all variation as random.  Specifically, these values will be considered as independent realizations of a random variable governed by one probability distribution.  Its mean characterizes the average arsenic concentration river-wide.

  1. Estimate the mean and variance assuming the underlying distribution is Normal.  Use unbiased estimators.
  2. Estimate the mean using the maximum likelihood estimator, assuming the underlying distribution is Lognormal.

2.    Environmental scientists are concerned about the 90th percentile of arsenic concentrations.  Accordingly, they obtain more independent random observations until they have 20 of them.

  1. The mean of the 20 observations is 29.3 and their standard deviation is 22.1 ppm.  Estimate their 90th percentile.
  2. The mean of the 20 logarithms is 3.12 and their standard deviation is 0.75.  Estimate the 90th percentile of the data (not of the logarithms!).

Extra credit: Estimate the variance of the data in problem 1 using the MLE for a Lognormal distribution.

Notes

The CDF of N(0, 1) at 1.282 equals 0.90.  The CDF of t(19) (Student's t with 19 degrees of freedom) at 1.328 equals 0.90.

The logarithms of the data in problem 1 are 2.0, 2.4, 3.6, and 5.4.

Scoring: The passing score is 92.  You must show details of your work to get credit for a correct answer or partial credit for an incorrect answer.

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