Problem 1
A newly developed test for a rare disease has the following features: if you do not suffer from the disease, the probability that you test positive (“false positive”) is 5%. However, if you do have the disease, the probability that the test fails to show it (“false negative”) is 10%. You took the test, and, unfortunately, you tested positive. The probability that you have the disease is:…………
Problem 2
You are going to play roulette. You first sit there and observe, and you notice that the last five times it came up “black.” Would you bet on “red” or on “black”?
Problem 3
A study of students’ grades in the United States showed that immigrants had, on average, a higher grade point average than US-born students. The conclusion was that Americans are not very smart, or at least do not work very hard, as compared with other nationalities. What do you think?
Problem 4
In order to estimate the average number of children in a family, a researcher sampled children in a school, and asked them how many siblings they had. The answer, plus one, was averaged over all children in the sample to provide the desired estimate. Is this a good estimate?
Problem 5
A contractor of small renovation projects submits bids and competes for contracts. He has noticed that he tends to lose money on the projects he runs. He has started wondering how he can be so systematically wrong in his estimates. Can you explain that?
Problem 6
Comment on the following.
[At a restaurant] ANN: I hate it. It’s just like I told you: they don’t make an effort anymore.
BARBARA: They?
ANN: Just taste it. It’s really bad food. Don’t you remember how it was the first time we were here?
BARBARA: Well, maybe you’re tired.
ANN: Do you like your dish?
BARBARA: Well, it isn’t bad. Maybe not as good as last time, but…
ANN: You see? They first make an effort to impress and lure us, and then they think that we’re anyway going to come back. No wonder that so many restaurants shut down after less than a year.
BARBARA: Well, I’m not sure that this restaurant is so new.
ANN: It isn’t?
BARBARA: I don’t think so. Jim mentioned it to me a long time ago, it’s only us who didn’t come here for so long.
ANN: So how did they know they should have impressed us the first time and how did they know it’s our second time now? Do you think the waiter was telling the chef, “Two sirloins at no. 14, but don’t worry about it, they’re here for the second time”?
Problem 7
Studies show a high correlation between years of education and annual income. Thus, argued your teacher, it’s good for you to study: the more you do, the more money you will make in the future. Is this conclusion warranted?
Problem 8
In a recent study, it was found that people who did not smoke at all had more visits to their doctors than people who smoked a little bit. One researcher claimed: “Apparently, smoking is just like consuming red wine – too much of it is dangerous, but a little bit is actually good for your health!” Do you accept this conclusion?
Problem 9
CHARLES: I don’t use a mobile phone anymore.
DANIEL: Really? Why?
CHARLES: Because it was found to be correlated with brain cancer.
DANIEL: Com’n, you can’t be serious. I asked an expert and they said that the effect is so small that it’s not worth thinking about.
CHARLES: As long as you have something to think with. Do as you please, but I’m not going to kill myself.
DANIEL: Fine, it’s your decision. But I tell you, the effects that were found were insignificant.
CHARLES: Insignificant? They were significant at the 5% level!
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