Quiz 9

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

An agency purports to have developed a 90% lower confidence limit (LCL) of the mean arsenic concentration in water as measured in 100 wells throughout southern Bangladesh.  These wells were randomly selected from all those used to supply water to southern Bangladesh (the "country").  The value of the LCL is 235 parts per billion (micrograms per liter).

The phrase "people in the country" below will mean "people who routinely drink untreated water from arsenic-contaminated wells like those surveyed by the agency."  (This apparently is about half the people living in Bangladesh.)

1.    Indicate which of the following statements are correct and which are incorrect.  Provide reasons for each.  Credit is given only for answers accompanied by reasons.

  1. At least 90% of the people in the country drink water with more than 235 ppb arsenic.
  2. There is a 90% chance that all the people in the country drink water with more than 235 ppb arsenic.
  3. The LCL procedure was chosen so that, regardless of the underlying distribution, there would be a 90% chance that the LCL would exceed the true mean concentration.
  4. The LCL procedure was chosen so that, regardless of the underlying distribution, there would be a 10% chance that the LCL would exceed the true mean concentration.
  5. The LCL procedure was chosen so that, regardless of the underlying distribution, there would be a 90% chance that the LCL would be less than 235 ppb.
  6. There is a 10% chance that the true mean equals 235 ppb.
  7. Any glass of water drawn at random from one of these 100 wells has a 90% chance of containing a mean arsenic concentration greater than 235 ppb.
  8. At least 90% of all arsenic-contaminated wells in the country have arsenic concentrations exceeding 235 ppb.

2.    To develop the LCL in problem 1, the agency assumed the sampling distribution of the mean would be Normal, but of unknown mean and standard deviation.  It also reports that the mean of the 100 concentrations was 320 ppb.   Compute the 90% upper confidence limit (UCL) of the mean.

Extra credit:  (a)  Compute the standard deviation of the data.  (The 10th percentile of Student's t distribution with 99 degrees of freedom is -1.29).  (b)  Using this information, explain why the UCL (as computed in problem 2) has likely been substantially underestimated.

Scoring: The passing score is 88.

NB: Arsenic is a "documented human carcinogen."  The US EPA maximum contaminant level (MCL) is 5 ppb, although the national drinking water standard is still 50 ppb.  Around the world, concentrations in water exceeding 10 to 50 ppb are considered unhealthful.  The "agency" in problem 1 and the numbers appearing on this quiz are made up but are consistent with available summary statistics (see below).

For more information on the arsenic problem in Bangladesh, see http://www.who.int/inf-fs/en/fact210.html (World Health Organization).
For some actual data, see http://www.kfunigraz.ac.at/fwiwww/aan/newsl2/contamin.html ("Asia Arsenic Network"), http://www.bicn.com/acic/resources/infobank/brac1.htm ("Arsenic Crisis Info Centre"), and http://ehpnet1.niehs.nih.gov/docs/1999/107p727-729tondel/abstract.html (Tondel et al., Environmental Health Perspectives Volume 107, Number 9, September 1999).

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This page was created 19 March 2001.