Using Statistical Methods For Water Quality Management.. Issues, Problems And Solutions 1st Edition Graham B. McBride - Solutions

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Why isn't the multiplier for the criterion of detection in Type B censoring, using 5% significance level, equal to 1.6449 (being the 95%ile of the unit normal distribution)? Derive the multipliers for this criterion and for the detection limit in Type B censoring at the 5% level (2.33 and 4.65).
Laboratory results below the limit of detection can be negative. Can you explain this? What should be done about it?
What information would be gained by performing a point-null hypothesis test on kappa? Would an equivalence test be useful?
Show that Equations (11.8) and (11.9) are equivalent.
accounts for chance-correction.
Explain how Table
Using the entries in Table 11.3, define the following terms: prevalence, overall proportion of agreement, sensitivity, specificity, false positive rate, false negative rate, overall proportion of disagreement, odds ratio.
Derive the p-value for the seasonal Kendall trend test when the test statistic is ZsK= 00.0707.
What are the occurrence formulae corresponding to Equation (10.10) for se- tups with 1, 2, or 4 dilution series?
Discuss the relative merits of evaluating Equation (10.2) via the recurrence relation [Equation (10.6)] versus the direct equation [Equation (10.7)].
Derive the recursion Equation (10.6), for the probability of getting r positive tubes in a set of R tubes, given n microorganisms to randomly distribute in that set.
Discuss how MPN values relate to the pattern of positive tubes in Table 10.3.
Show that the two forms shown in Equation (9.25) are identical.
Derive the beta-binomial conditional dose-response relationship for variable pathogen survival probability [Equation (9.23)].
Derive the exact dose-response relationship for variable pathogen survival prob- ability and a random distribution of pathogens [Equation (9.20)].
Derive the simple exponential dose-response model for constant pathogen sur- vival probability and random distribution of pathogens [Equation (9.17), which is also Equation (9.1)].
Derive the hockey stick equations [Equations (9.9) - (9.11)].
What is the sum of Equation (9.7) over all nonzero values of the integer k?
What is the probability of getting a negative result when sampling a contami- nated water in which pathogens are present and distributed randomly in a proportion P of samples?
What is the value of the mode of the shellfish meal size distribution shown in Figure 9.2?
In Section 9.1.2 we noted that it may be better to use the mode rather than the median when constructing a distribution of ingestion or inhalation rates, given three statistics for these rates (in mL per hour): minimum = 10, median = 50, maximum = 100. Why is this?
Discuss the difference in health risks predicted by QMRA based on a individ- ual exposed one million times, versus 1000 individuals exposed 1000 times. Hint: contrast cases for exposure to a very infective versus a much less infective pathogen.
Show that the p-value for the proof of hazard approach [Equation (8.4)] may also be written in terms of the incomplete beta function ratio, ph = Ix (en+1, n-en), and show that this is also the Confidence of Compliance that results when taking a Be(1, 0) prior distribution.
Explain why a proof of safety approach means that the proportion of ex- ceedances among samples is always lower than that contemplated for the whole population.
Explain why a proof of hazard approach means that the proportion of ex- ceedances among samples is always higher than that contemplated for the whole population.
In Figure 8.2(b) the estimated 95%ile declines as the number of observations increases. Can you explain why this happens?
Discuss the Hazen estimate of a 95% ile when 10 observations are available.
In Section 8,1 we observed that requiring a wastewater treatment plant to meet its historical long-term 95%ile for ammoniacal-nitrogen would mean that the plant would be found to be in breach of its consent about half the time. Why is this? What would be the implications of requiring the plant to
When is the question "How many samples should be taken to assess compliance with a water quality standard?" solely a statistical issue?
Discuss the notion that considerable care is needed if null hypothesis tests are to be used for drinking-water standards, and that it would be best if they were not?
A precautionary proof of safety compliance rule can be based on one-sided confidence limits. So can permissive proof of hazard rule. How can this be done? How do these approaches compare with a face-value stance?
Discuss the merits of having "running" compliance assessment periods, versus adjacent periods.
Discuss the merits and disadvantages of setting maximum standards for receiv- ing waters.
Discuss the merits and disadvantages of setting maximum standards for drink- ing waters.
Discuss the merits and disadvantages of setting maximum standards for efflu- ents.
Fines or enforcement notices are seldom imposed for effluent consents that exceed a maximum limit (a 100%ile). Why is this?
Consider the case where an effluent consent contains only receiving water stan- dards, not effluent standards. For example, "...the consent holder shall not cause the dissolved oxygen in the stream anywhere downstream of the point of discharge to fall below 80% saturation." Is this good practice?
Discuss the merits of including receiving water standards in effluent consents.
Pathogenic microorganisms such as oocysts of Cryptosporidium parvum may pass through a poorly performing filtration unit in a water treatment plant. Contin- uous monitoring of such microorganisms is therefore desirable, but it is not always feasible even frequent assays may be impractical and/or
Should stormwater standards be stated in permits or given a more general au- thorization, such as in a water quality management plan?
Discuss the issues arising when "minor" wastewater discharges are granted a "general authorization" (e.g., in a watershed management plan).
What happens to the shape of the "chimney on the teepee" in Figure 5.12 as the standard error on the y-axis increases?
Show that the top of the detection region for the inequivalence hypothesis(variance unknown) in Figure 5.12 is where the standard error equals 20/t1-a,f
Show that the detection probability for the inferiority hypothesis, with variance unknown, is Pr(t'
Discuss the behavior of tests of the inequivalence hypothesis as the width of the equivalence interval → 0.
Show that as the width of the equivalence interval 0, the decision rule presented in Table 5.9 for the test of the equivalence hypothesis with known variance becomes identical to the rule for the point-null hypothesis test with known variance (in Table 5.7). Does this convergence of decision rules
Derive a recursive algorithm for the hypergeometric distribution.
Derive a recursive algorithm for the negative binomial distribution.
Show that the negative binomial pmf b* (n; k,0) = ()0k(1 - 0)n-k for n = k,k+1, k+2,...can also be expressed as b(s;k, μ) = (*+-1)()k()for s = 0, 1, 2,..., where u is the distribution's mean.
Show that, for the negative binomial distribution, B* (2k - 1; k, ) =
What role does the parameter play in the geometric distribution? Can this be generalized to the negative binomial distribution?
Show that the negative binomial and binomial distributions' pmfs are related by b*(n; k, 0) = b(k; n,0).
Derive a recursive algorithm for Poisson probabilities by noting that p(k + 1, μ) = p(k; μ).
Show that the binomial pmf b(k; n, 0) can be calculated directly from exp[ln Г(n +1) - ln I(k +1) In I(n- k+ 1) + kln(0) + (n- k) In(1- 0)].Can the CDF [B(n; k, 0)] be calculated using this formula?
Derive the recursive algorithm given in Section 6.2.5 for the calculation of binomial probabilities. Hint: First demonstrate that b(k+1; n,0) (k1)-b(k;n,0).What is the advantage of this algorithm over the direct algorithm presented in Problem 6.10?
Compare the normal approximation to the binomial pmf (see page 143) to the exact binomial value (use n = 1000,0 = 0.1 and e = 90). Also, compare the Poisson approximation to the binomial pmf (also on page 143) to the exact binomial value(use n = 1000,0 = 0.01 and e = 3).
Show for the binomial distribution that B(k; n, 0) = B(n- k;n, 0) only if the distribution is symmetric-B(k; n,0) is the probability of getting up to k "successes"in n random dichotomous trials each of which has the probability of success, and B(k; n, 0) is the probability of getting k or more
Show that the beta distribution CDF, F(x;a, ẞ), is the incomplete beta function ratio I, (a, B), where I, (a, B) S Ea-1(1 - 3)3-1 dE :a, B > 0. B(a,B) J0
Show that the beta distribution pmf, F(x;a, ), can be evaluated from exp{In[F(a)] In[T(a)] - In[r(3)] + (a-1) In(x) + (3-1) In(1-x)}.
Show that the gamma distribution CDF, F(x;a, 6), is the incomplete gamma function ratio P(a, 1/3), where P(a, x) = (a) Joe-a-1 d.
and also 1.0.
to the ratio of any two percentiles of normal and (two-parameter) lognormal distributions. Use the results to calculate the ratio of the 99%ile to the 95%ile for normal and two-parameter lognormal distri- butions with coefficient of variation =
Generalize the results obtained in Problem
Safe shellfish harvesting criteria (USDHHS 1993 [322]) require that the median concentration of fecal coliforms in the shellfish-growing water should not exceed 14 per 100 mL, nor should the 90%ile exceed 43 per 100 mL (based on 15 samples, a "systematic random sampling strategy" and a particular
Show that the CDF of the three-parameter lognormal distribution (Equation 6.3) is given by F(x, y, Oy, T) = G (In( In(-)-y unit normal distribution. ), where G(r) is the CDF of the
Explain why the trios (ni, de, dp); (nj, de, np) and (d;, de, np) don't appear in Figure 5.13.
Explain why the duo (ni, de) doesn't appear in Figures 5.12(a) and 5.12(b).
In Problem 5.2 the test statistic is larger when the data are treated as paired, compared to treating them as two independent samples. This will result in a smaller p-value and it therefore seems that pairing has made the test more "powerful." It is often held that pairing always makes a test more
for the paired-sample case?
What modifications should be made to Table
What is the limiting form of Figure 5.8 as n → 00?
What is the limiting form of Figure 5.5 as n →∞? Hint: Ho is true only at▲ = 0; K is true everywhere else.
Consider a single sample case, where we wish to make a test on the mean () of a normal population, with unknown variance. We ask, is it zero? From Section 5.2, the point-null hypothesis Ho postulates that =o = 0. From Tables 5.4 and 5.7 we reject this hypothesis if |T|> t- where T = Show that this
Repeat the Houhou Stream example given in Section 5.1.3 treating the data as paired samples. Discuss the meaning of the results.
In Problem 4,1 on swimmers' health a point-null hypothesis was not rejected in a 5% level test, because the data gave rise to p = 0.059. Discuss the possible outcomes of this study were the study to have had double the number of swimmers participating.
Using the drinking-water example in Section 4.4, show that the posterior hazard probability aftera sequence of n positive laboratory testresults is Po(1-6) Po(1-B)n+(1-Po)an6 where a is the test's false positive rate, ẞ is its false negative rate, and Po is the prior hazard probability adopted
What is the relationship between laboratory false positive and false negative rates, and the Type I and Type II error rates (a and 3) for hypothesis tests, as shown on Table 4.1?
[Difficult problem] Show that the p-value for the two-stage sampling regime referred to in Figures 4.3 and 4.5 (with 20 observations and possibly a further 20 observations if the first set fails to attain statistical significance) is p = 2B(5; 20, )+ b(6; 20, ) B(7; 20,)+ {( 20,) [B (13-i; 20, )+B
Similarly, show that the p-value for the single-stage sampling regime (with 40 observations) referred to in Figure 4.5 is 2B(13; 40, 0).
Discuss the special negative binomial case [B* (2k 1; k, ) =] demon- strated in Problem 6.14 in terms of the probabilities shown on Figure 4.4.
Show that the p-value for a single-stage sampling regime (with 20 observations) referred to in Figure 4.4 is 21* (20; 5, ).
is B(5; 20, 1)+B(15; 20, 1) = 2B(5; 20, }) As demonstrated in chapter 2, B(k; n, 0) is the cumulative binomial probability of getting up to k successes in n random trials each with a probability of success; similarly, B(k; n, ) is the probability of getting k or more successes.) Hint: Use results
Show that the p-value for Figure
A study has been of the effect of animal faecal residues in recreational water on the health of those swimming in that water (Calderon et al. 1991 [39]). The study sought to see if there was an association between the concentration of fecal indicators and pathogens and swimmers' health. It did so
If you were implementing a calculation procedure for one-sided confidence(or tolerance) limits on normal percentiles some provision would have to be made for the k values for sample sizes not reported on (Table A.4.2). How could this difficulty be resolved?
When calculating one-sided confidence (or tolerance) intervals from the formula X + kS, the value of k (Table A.4.2) for 50% confidence for any percentile(except the median) differs from the equivalent percentile of the unit normal distribution. For example, when estimating the 95%ile with n = 20
Confidence intervals and prediction intervals for a linear regression have a waist, where the distance between the between the upper and lower limit is at a minimum. Where does this occur?
What happens to the width of the confidence intervals for paired and unpaired samples in the example given on page 65 if the eighth X value is changed from 12 to 28 parts per billion of DRP?
Is the standard error for n paired samples always less than the standard error for two independent samples each of size n?
Show that the standard error of the mean difference of n paired samples is 2 Se SEXY n(n-1) - nXY],
Show that the standard error for the difference in means between two independentsamples each of size n is Se,-y = √n(n-1( X? + Σ Y? - n (X2 + Ỹ2)].
Derive one-sided confidence interval equations for the difference between two independent normal means, where each sample is of size n.
Why is a nonparametric confidence interval for a central tendency statistic based on the median, not the mean?
Consider the set of 10 E. coli data for beach water, expressed as numbers per 100 mL of river water: 7, 15, 22, 432, 122, 52, 25, 14, 43, 33. What are the appropriate 95% confidence limits for the central of the distribution these data have been drawn from?
Derive Equation (3.2) from Equation (3.1).
In the section on interpreting confidence intervals in a Bayesian manner (page 58), it is observed that the prior distribution invoked postulates that "large" positive values of lake chlorophyll a are more likely than are small values. Explain why this is so, and discuss its implications for the
Discuss the meaning of the coefficients in the logistic model [Equation (2.25)].
Forty one pairs of data (X and Y) (in parts per billion) are obtained for two methods of estimating chlorophylla, with X = 2.07146 and Y = 2.81805 and the following regressions: Y = 0.86947X + 1.0170 and X = 0.94901Y - 0.60288.Which regression equation should be used? What are its main features?
We have the following two groups of data for the concentration of the bacterium E. coli per 100 mL of water on each of 11 separate days: site 1-100, 82, 1198, 336, 5, 1071, 156, 31, 1120, 156, and 5 (average = 387.27); site 2-50, 20, 146, 738, 98, 1401, 98, 6488, 970, 98, and 50 (average = 923.36).
Does the sample geometric mean overestimate or underestimate the geometric mean of a lognormal distribution?.

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