Variação Amostral

Alguns exemplos onde a variação está presente no dado podem ser encontrados em Landim (1997).

1. Funç ao pulmonar em pacientes.

   Maximal static inspiratory pressure (PImax) is an index of respiratory muscle strength. The following data show the age (years) and a measure of PImax (cm H$_2$O) for 25 cystic fibrosis patients.

   
   1. Do all cystic fibrosis patients have the same PImax value?

   2. Assuming age does not affect PImax, what is a ``typical'' PImax value for systic fibrosis patients?

   3. How large is the variability about this typical value?

   4. Is the assumption that age does not affect PImax consistent with the data?

   5. If age does affect PImax, how would you describe the ``typical'' PImax value and variability?

2. Fat and protein content in milk.

   Scientists measured the fat and protein content in milk samples from 10 grey seals.

   
   1. Are the percentages exactly the same from one animal to the next?

   2. Based on this sample of 10 seals, the scientists estimated the fat content in milk from grey seals to be 53.7%. If they now collect milk samples from a different 10 seals, would you expect their estimate to be 53.7%?

   3. How does the sample size influence your answer?

   4. What if they took another set of samples from the same 10 seals? Would you expect to get the same estimate in that case?

3. Sunflower plant experiment.

   The table below shows the results of an experiment where the researchers were interested in the role of nitrogen, magnesium and certain micro-nutrients in the metabolism of sunflowers. Twelve sunflower plants were grown in water culture for nine weeks, under uniform conditions of lighting and temperature. Each plant was in a separate pot with one type of culture solution. There were four batches of three plants: A, B, C, D. The three plants in Batch A were grown in a complete culture solution. Those in Batch B had the same culture solution except that it lacked magnesium. The solution for Batch C lacked only nitrogen and that for Batch D lacked only th micronutrients (boron, manganese, zinc and copper). At the end of the experiment each plant was chopped up, dried in an oven and then weighed, giving the results in the table.

   
   1. What can we conclude about the effect of the different solutions on the plants?

   2. How confident are we of our conclusions?

   3. How could the experiment have been designed such that we would be more certain about the effect of the differing culture solutions?

   4. If you were to repeat the experiment with a very large number, say 1000, plants in each batch, where would you expect the averages to lie? For example, in Batch B I'd think that the average will most likely lie somewhere between say 50 and 100. I'd be very surprised to find that it was 1000. What range of plausible values would you give for the other batches?

   5. Does everyone have the same conclusions and ranges of plausible values? Why not?

   6. Wouldn't it be useful to have a more systematic way of describing the results of the experiment and of quantifying our uncertainty?

   7. Are we happy with the way the experiment was set up? Are there other questions that we might want to ask? How could we design the experiment so that we would be more certain about the long-term average heights that future plants would grow to under the different cultures?

4. An early biomedical trial

   One of the earliest recorded comparative biomedical trials (c. 597 B.C.) is a nutritional experiment from the Book of Daniel in the bible. The background is that Daniel was held hostage, and disapproved of the rich food, wine, and meat served. In order to convince the guard in charge that he and his three friends should be given a simple diet of vegetables, Daniel asked for a 10-day trial. He promised to eat the assigned diet if those on the vegetable diet weren't as healthy as the others.

   Then Daniel said to the guard $\ldots$ ``Submit us to this test for ten days. Give us only vegetables to eat and water to drink; then compare our looks with those of the young men who have lived on the food assigned by the king, and be guided in your treatment of us by what you see.'' The guard listened to what they said and testedn them for ten days. At the end of the ten days they looked healthier and were better nourished than all the young men who had lived on the food assigned to them by the king. So the guard took away the assignment of food and the wine they were to drink and gave them only vegetables.

     Chapter 1, Verses 12-15, New English Bible Version.

   
   
   
   

   What do you think of Daniel's experiment? That is, are you convinced that the vegetable diet is better? Why or why not?