Because the previous editions of the math and science threads have been dead too long for any hope of resuscitation, I'm giving birth to a new generation for the math, science and technology lovers of this generation. Feel free to discuss anything regarding the unbelievably general topics of math, science and technology here.
SciMaTech is a shorthand version of writing science, math and technology if that wasn't obvious.
Yay Sal! I've been wanting the math thread to come back and now it kinda is back!
Here's a math puzzle...
Create the numbers 1-80 using four and only four fours and any mathematical devices except for pretty much logs, sigmas, and trig functions. I guess infinity as an idea or whatever can't be used either since technically the infinite number of square roots put on any fraction (such as .4 or .4 repeating with the line over it, 4/9) will always equal one, and that would only take one four to make 1, which shouldn't happen.
To give some examples, "2" can be created by 4/4+4/4. "3" can be created by (4/(radical)4)+4/4. "64" can be created by "4^(4+(4/4)). Those are some easy ones, though. I've successfully done all the numbers 1-80 except for 37, 71 77, 78, and 79.
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Speaking of four, for a given integer n in which f(n) is the number of letters in the word, what will an infinite number of applications of f(f(f(f(f(...)n) always lead to?
4 x 1 = 4 4 x 2 = 8 4 x 3 = 12 4 x 4 = 16 4 x 5 = 20 4 x 6 = 24 4 x 7 = 28 4 x 8 = 32 4 x 9 = 36 4 x 10 = 40 4 x 11 = 44 4 x 12 = 48 4 x 13 = 52 4 x 14 = 56 4 x 15 = 60 4 x 16 = 64 4 x 17 = 68 4 x 18 = 72 4 x 19 = 76 4 x 20 = 80
Bill Nye the Science Guy lol. Seriously though, Harvard recently put out a great series dedicated to teaching people how to create iOS apps using Objectice-C. I don't know how to link from the App Store, but it'll provide days of entertainment and it teaches something useful.
So, as you are probably aware, its pretty easy to poke holes in just about any argument or theory, no matter the scientific credibility. Most of that is just nitpicking and doesn't really amount to much, but its also important to know how to be a good skeptic. Also, its more fun (that is, easier) than coming up with a constructive solution. So! I'll put two examples (for now) in front of you guys, the challenge being to find the biggest (and most likely) problems! You are encouraged to ask clarifying questions.
1) In a study that examines how stress effects vole population sizes, I measure various hormone levels from many specimens taken from a population as it reached different population densities. A statistical test found a significant difference between the stress levels of high density populations and low density populations. The sample size was large enough.
2) A case-control study measuring the effects of alcoholism on ovarian cancer was conducted at Massachusetts General Hospital. The control group consisted of patients with ovarian cancer and the control group consisted of a random sample of all emergency room patients without any cancer. An interview determined the proportion of each group that were very heavy drinkers (for a significant period of time during the past 10 years), and found there was no significant relationship between heavy drinking and ovarian cancer.
Have fun! (Try not to focus on the vagueness of the situations TOO much, these are not actual studies as far as I know so all information is made up)
^ The vagueness is a bit of an issue, at least for me.
For the first one, is it an observational study, or are we talking about a controlled environment? What factors are being controlled? I don't know much about voles (or animal behavior in general), but I would assume that environmental differences would cause differences in population densities. If that's the case, that'd be a confounding variable. To add on to that point, from a natural selection standpoint, environmental differences can lead to genetic differences (i.e., hormone levels are higher in general for one population simply because that population is more genetically inclined to produce more stress-indicating hormones) Moreover, how are the physical boundaries of the areas from which the densities are calculated determined? And in general, the level at which statistical significance is determined is often quite arbitrary.
For the second one, there could be a response bias in the survey. Heavy drinkers may not necessarily want to admit they they're heavy drinkers. Further, I'm not really understanding the groups, in part because you called them both control groups. I'm assuming the actual control group is the group without ovarian cancer, and we are comparing the ratios of alcoholics between the two groups? If that's the case, I guess another question I have is why the second group has to consist of emergency room patients. Why can't it just be healthy individuals? It's possible that the the emergency room may contain a higher prevalence of alcoholism relative to the rest of the population, as there may be people hospitalized for alcohol-related issues.
The control group consisted of patients with ovarian cancer and the control group consisted of a random sample of all emergency room patients without any cancer.
There's your problem. There're two control groups, though I'm assuming that was (hopefully) a typo.
Anyway, with the second one, any type of interview-based study is bound to yield results embedded in human error, the most vulgar being the ignorance of the interviewee. As my physics professor exclaims every time we have a lab report, "Don't ask your parents for help! Humans don't know nothin'!"
There's your problem. There're two control groups, though I'm assuming that was (hopefully) a typo
Yup, as Ray pointed out, the group without cancer is the actual control group. My Bad.
any type of interview-based study is bound to yield results embedded in human error
Two problems with this: 1) That would basically discredit all survey based studies, which is quite a few studies. Surveys can be created to reduce bias, and they are a (currently) necessary tool for a lot of research 2) In terms of human error... I would think most people would know their own alcohol consumption habits. Now, whether or not they would be truthful in reporting this knowledge... There's a problem (Which ray also pointed out).
Why can't it just be healthy individuals? It's possible that the the emergency room may contain a higher prevalence of alcoholism relative to the rest of the population, as there may be people hospitalized for alcohol-related issues.
Well, which group of healthy individuals? That's why many epidemilogical studies use control groups from emergency rooms- it's just very practical. But, your point is still valid- emergency room patients probably have a higher incidence of alcoholism than the general population. Also, emergency room patients in a large hospital in a huge city (like Mass general) would be from all over the world, whereas those in the cancer ward are probably much more local.
Both of you, however, missed the biggest problem: not everyone has ovaries. Taking a random sample from a population that includes men (without excluding the men) as the control group would almost certainly invalidate the entire study.
but I would assume that environmental differences would cause differences in population densities.
Well, it might not have been clear, but in my first sentence I said that the samples would be taken from one population as it reached different population sizes, thus controlling for many of the confounding variables that you mentioned. Of course, not all variables can be controlled for, but science isn't a perfect science.
For this one, I was more thinking along the lines that the actual test itself would probably be a very traumatic experience for the voles (I didn't state this, but these tests typically involve taking a blood sample), thus inflating the stress levels. Maybe it wasn't such a good example.
Care to post any of your favorite riddles from those sites? I perused a bit, and they seem rather challenging to say the least.
So, this study is kind of a huge deal (and obviously very controversial). It has gotten quite a bit of attention since it was published in 1999, and there is a lot of debate out on the nets about it. So.
Basically, what they did was randomly assign a bunch of admitted cardiac patients to either the prayer group or the non prayer group. Prayer consisted of a third party praying for the health and speedy recover of the admitted patient. None of the patients even knew they were part of the study, and were not informed that were/were not being prayer for. The pray-er was only given the first name of the patient.
Then, they created an index by assigning point values to different negative complications/events that could happen while admitted, with worse things (IE cardiac arrest/death) being attributed more points than less serious events, like a need for antibiotics. Thus, each patient would accrue points throughout their stay, and be given a final score, with a higher score indicating greater harm/suffering/etc. This index was created by the Mid America Heart institute for use in this study.
And, lo and behold, the prayer group had significantly lower scores than the non prayer group, with a p value of 0.04! WOW! The authors concluded that prayer should be used alongside medicine. Please note that this is a peer reviewed paper published in a typically credible journal, the archives of internal medicine.
So... did they do anything wrong? (the answer is "Yes", by the way)
Both of you, however, missed the biggest problem: not everyone has ovaries. Taking a random sample from a population that includes men (without excluding the men) as the control group would almost certainly invalidate the entire study.
I thought all subjects being women was sorta implied, lol. I mean, if that were a real study, there's no way it'd pass the peer-review process, since that'd egregiously skew the results. But it was fun to think about nonetheless.
