Chapter CR, Problem 6E

Lake source cooling Since 2000, Cornell University has used a lake source cooling plant that circulates water from Cayuga Lake through a heat transfer system to chill water for use on campus in refrigeration and air conditioning. Supporters of this method cite reduced carbon emissions and energy costs. Critics fear that higher levels of chlorophyll may damage the ecology of the lake. Cornell monitors the lake at seven test sites near the discharge pipe and at a control site farther away. A report found a statistically significant (at α = 0.05) higher mean chlorophyll level relative to the control site at site #7—one of the monitoring sites.

1. a) A long-time critic of lake source cooling is quoted by the Ithaca Journal saying, “The finding shows that the lake source cooling is impacting the … lake.” Which of these comments about this statement is most appropriate?
   1. i. He’s correct: That’s what a significant finding means.
   2. ii. He’s wrong: The finding is probably due to chance.
   3. iii. He’s correct: Comparing to a control site makes this an experiment.
   4. iv. He’s wrong: This is really an observational study, not an experiment. We can conclude that there is a discernible difference, but we cannot infer a cause.
   5. v. He’s correct: There are no other plausible explanations, so lake source cooling must be the cause.
2. b) Cornell appears to have performed a hypothesis test. What, specifically, was the null hypothesis of this test?
   1. i. H₀: Chlorophyll levels have not changed in the 8 years of lake source cooling.
   2. ii. H₀: μ_{TestSite#7Chlorophyll} − μ_{ControlSiteChlorophyl}=0
   3. iii. H₀: ӯ_{TestSite#7Chlorophyll} − ӯ_{ControlSiteChlorophyl} = 0
   4. iv. H₀: μ_{TestSite#7Chlorophyll} = 0
   5. v. None of the above
3. c) The alternative hypothesis of Cornell’s test was most likely
   1. i. one-sided, testing for higher chlorophyll levels at the test site.
   2. ii. one-sided, testing for lower chlorophyll levels at the test site.
   3. iii. two-sided.
   4. iv. rejected.
   5. v. irrelevant.
4. d) An associate dean of the university is quoted by the Journal pointing out that 28 tests were actually performed and that the one cited by the critic is the only one that showed a statistically significant result. He says, “Scientists expect that pure chance will result in a statistically significant result in every 1 in 20 comparisons.” In proper statistical language, what is the dean implying?
   1. i. This result may have been a sampling error.
   2. ii. This result may have been due to response bias.
   3. iii. This result may have been a Type I error.
   4. iv. This result may have been a Type II error.
   5. v. This result may have been a Type III error.
5. e) You’ve been called in to consult as a statistics expert. You are asked to comment on whether the hypothesis test is appropriate. What would you like to know or see about the data?
   1. i. A time plot showing the chlorophyll levels at the site for the past 8 years to check for linearity.
   2. ii. The correlation of chlorophyll levels at test site #7 and at the control site.
   3. iii. A histogram of the chlorophyll measurements at test site #7 and at the control site to check for outliers, skewness, or bimodality.
   4. iv. An independent assessment of the chlorophyll levels performed by New York State environmental officials.
   5. v. Boxplots of the chlorophyll levels at all seven of the test sites to check that the variation is approximately equal.
6. f) An environmental expert has noted that chlorophyll levels are ordinarily quite variable and has questioned whether the effect size of this finding is large enough to be of environmental concern. Statistically, what is he saying?
   1. i. The observed difference isn’t really significant.
   2. ii. This was probably a Type I error.
   3. iii. The test isn’t powerful enough.
   4. iv. Although the difference is significant, it may not be meaningful.
   5. v. Although the difference is profound, it may be too subtle.