Has the composition of Earth's atmosphere changed over time?  To study this question, geologists Robert Berner and Gary Landis (1988) examine the composition of gas bubbles in ancient pieces of amber (hardened tree resin preserved in sedimentary rocks).

To determine the composition of the gas bubbles, they crush the amber in a vacuum and analyze the released gases with "time-resolved quadruple mass spectromety"   (Berner and Landis, 1988, p. 1406).  After arguing that the air in the bubbles is not contaminated by modern air, Berner and Landis (1988) present the percentages of nitrogen and carbon dioxide plus oxygen in nine gas bubbles in amber from the Upper Cretaceous age (about 75 to 95 million years ago).  These data are shown below.  In the sample labels, the Roman numerals correspond to the piece of amber that is crushed (there are three pieces), and the letter corresponds to the gas bubble within the amber that is analyzed.

                                                                                                                Sample Label

Gas	IA	IB	IIA	IIB	IIC	IID	IIIA	IIIB	IIIC
N2	63.4	65.0	64.4	63.3	54.8	64.5	60.8	49.1	51.0
CO2 + O2	33.5	30.5	28.3	28.4	32.3	25.5	36.6	27.8	25.5

Berner and Landis (1988) argue that the carbon dioxide is respired oxygen from trapped microorganisms, so that the original levels of oxygen in the amber equal the CO₂ + O₂ percentages.  Thus, they claim these are percentages of the two major gases from nine samples of ancient air.

This is a study involving inference for individual means.  The data are in the file ancientair (click here to download).

Questions

1a) (not handed in) Examine the normal quantile plots of nitrogen and oxygen to see if their distributions can be described roughly with normal curves.  Recall, to get a normal probability plot, first run Analyze - Distribution with the variable of interest, click on the red arrow next to the variable name, and select Normal Quantile Plot.   You don't have to turn anything in for this part of the questions, just make the plots to get in the good habit of checking assumptions before doing significance tests.  You'll also use Analyze-Distribution to do the significance test.  If the normal curves reasonably describe the data, the hypothesis tests are OK to run.  Otherwise, you have to use methods not covered in this course.  We'll go ahead and run the tests regardless, just for practice.

1.  Modern air is known to contain 78.1% nitrogen and 20.9% oxygen.  Is there evidence that the percentage of nitrogen in ancient air differed from the percentage in modern air?  Use a two-sided  t-test, since we are looking for any differences from the modern percentages.  Write on your report your hypotheses, the value of the test statistic (show the numerator and denominator that go into the test statistic), the p-value, and your conclusion.

To do a t-test in JMP for a single mean, first run Analyze - Distribution on the variable of interest.  Then, click on the arrow next to the variable name, and select Test Mean.  Enter the hypothesized value of the mean in the first box.  Leave the second box empty.  Click OK.   JMP reports three p-values at the bottom of the output:  1) Prob > |t|, which is the p-value when the alternative is two-sided; 2) Prob > t. which is the p-value when the alternative has a >  sign in it; and, 3) Prob < t, which is the p-value when the alternative has a < sign in it.  Choose the one that matches your alternative hypothesis.

2.  Is there evidence that the percentage of oxygen in ancient air differed from the percentage in modern air?  You can report simply the value of the test statistic, the p-value, and your conclusion.

3.  Form a 95% confidence interval for the percentage of oxygen in ancient air.  For simplicity, use 2 as the multiplier.  You can get the relevant mean and standard error from the JMP output.  Write your 95% CI on the report, and explain in one sentence what this interval tells you about the percentage of oxygen in ancient air.

COMMENTS ON THIS PROBLEM.

It is not universally accepted by geologists that the gas bubbles represent samples of air from ancient times.  This question can be answered only by experts in the field.  However, the appropriateness of t-tests for these data can be criticized from a statistical point of view. Two criticisms include: 1) (serious criticism) the observations are not independent, since the samples come from the same rock; and, 2) (less serious criticism) the  N₂ values are not symmetric around the mean, which could make the tests inaccurate with a small sample size. The second criticism doesn't bother me too much, since the sample average is so many SEs away from 78.1% that there's no doubt the chance of seeing a sample average as or more extreme than the value in the data is very small.  
   
Reference:  Berner, R. A. and Landis, G. P. (1988) "Gas Bubbles in Fossil Amber as Possible Indicators of the Major Gas Composition of Ancient
Air." Science 239, pp. 1406--1409.

Is caffeine dependence real?

The subjects are eleven people diagnosed as being dependent on caffeine.  During one time period, these people were barred from coffee, colas, and other substances containing caffeine and instead took capsules containing their normal caffeine intake.  During a different time period, they took placebo capsules with no caffeine.  The order of the time periods in which the subjects took caffeine and placebos was randomized.   The subjects, pill administrators, and testers did not know when they got each pill.

Subjects were assessed on the Beck Depression Inventory, which is a psychological test that measures depression.  Higher scores on the test mean the subject shows more symptoms of depression.   Additionally, subjects were asked to press a button 200 times as quickly as possible, and their number of presses per minute was measured.   The researchers are interested in whether being deprived of caffeine affects either of these outcomes.

This is a matched pairs study, because comparisons of the treatments are made on the same person.  The data are in the file caffeine (click here).

Questions

Make new columns for the differences in depression scores and in beats.  For both differences, take (caffeine score - placebo score) as the ordering.   To input the differences, you'll have to edit the columns and use a Formula.  Call over a TA if you have trouble doing this.  

4a (not handed in) Examine the distribution of depression score differences (caffeine - placebo).  Does a normal curve seem like a reasonable description of the differences?  You don't have to turn anything in for this part, just make the plots to check assumptions.  If the normal curve seems like a reasonable fit, you can use the t-test approach.  Otherwise, you have to use other methods that we have not covered in this course.

4.  Test the hypothesis that there is a difference in the average depression score of people on the caffeine pill and people on the placebo.  Write on your lab report your hypotheses, the value of the test statistic (show the numerator and denominator that go into the test statistic), the p-value, and your conclusions.

To run the t-test, follow the same directions as in the ancient air problem.  Use 0 as the hypothesized mean (for no difference).

5a (not handed in) Examine the distribution of differences in beats (caffeine - placebo).  Does a normal curve seem like a reasonable description of the differences?  You don't have to turn anything in for this part, just make the plots to check assumptions.  If the normal curve seems like a reasonable fit, you can use the t-test approach.  Otherwise, you have to use other methods that we have not covered in this course.

5.  Test the hypothesis that there is a difference in the average beats of people on the caffeine pill and people on the placebo.  You can simply report the p-value and the conclusions you reach about the effects of caffeine on beats.
 

Reference:
Moore, D.  The Basic Practice of Statistics.  New York:  W.H. Freeman, 2000, p. 382.

Subliminal Messages (you will get this problem) and Their Effects on Math Test Scores (you will get this problem)

A subliminal message is below our threshold of awareness but may influence our behavior.  Can subliminal messages affect the way students learn math? A group of students who had failed the mathematics part of the City of New York Skills Assessment Test agreed to participate in a study of this question.  The data were originally collected in a study by John Hudesman, and the study is described in Moore (2000, p. 400).

All students received a daily subliminal message flashed on a screen too rapidly to be read consciously.  The students were randomly assigned to receive one of two messages. The treatment group received the message, "Each day I am getting better in math."  The control group received the neutral message, "People are walking on the street."  All students in both groups took a pre-test, went to a summer math skills program, and then took a post-test.

This is a study involving inferences for the difference in means of separate groups.  It's not matched pairs because there are two separate groups: the students who got the subliminal message, and the students who got the neutral message.  The data for the students' test scores are in the file subliminal (click here).  People in the subliminal group have the code "T", and people in the neutral message group have the code "C".

Questions:

6a) (not handed in)  In this problem, the outcome variable is the improvement in test scores.   For each group, examine the distribution of improvement scores.  Do normal curves appear reasonable descriptions of the distributions of improvement scores in each group?  You can get both normal curves on one plot by using Analyze - Fit Y by X.  Put the continuous variable in the Y-box and the group variable in the X-box.  After running it, click on the red arrow next to the "Oneway analysis...", and select Normal Quantile Plot - Plot Actual by Quantile.   If the data in both groups roughly follow normal curves, we can proceed with the significance test.  Otherwise, you use methods that we have not learned in this course.

6.  Is there evidence of a difference in the average improvement in test scores (post-test score - pre-test score) for the subliminal and neutral message groups?  Write your hypotheses, the value of the test statistic (show the numerator and denominator that go into the test statistic), the p-value, and your conclusions.

To run a hypothesis test for the difference of two means in JMP,  use  Analyze - Fit Y by X, inputting the continuous variable in the Y-box and the group variable in the X-box.  After running it, go to the red arrow next to the "Oneway analysis..."  Then select Unequal Variances. The output at the very bottom is the test.  The entry under "t-Test" is the value of the test statistic.  The entry under "Prob > F" is the p-value for the two-sided alternative hypothesis.  

7.  Give a 95% confidence interval for the difference in average improvements between the subliminal and neutral groups.  Use 2 as the multiplier, and pick off the appropriate values for the means and standard errors by clicking on the red arrow next to "Oneway analysis..." and selecting Means and Std Dev.  Explain in one sentence what this confidence interval tells you about the effectiveness of the subliminal message versus the neutral message.

COMMENTS ON THIS PROBLEM:

These conclusions are valid for the subject material, message, and student populations in this study. However, they may not generalize to other subject material, messages, or other populations.  Additional studies involving other subject material, other messages, and other populations are needed before we can feel  secure with broad generalizations.

Reference:
Moore, D. The Basic Practice of Statistics. New York: W.H. Freeman and Company, 2000.