Bayes's theorem from Tversky and Kahneman in Michael Lewis's The Undoing Project

Question

Michael Lewis's book "The Undoing Project" is concerned with the (mathematical) psychologists Daniel Kahneman and Amos Tversky. (Kahneman won the 2002 Nobel Prize; Tversky died in 1996.) On page 157, this question is quoted:

The mean IQ of the population of eighth graders in a city is known to be 100. You have selected a random sample of 50 children for a study of educational achievement. The first child tested has an IQ of 150. What do you expect the mean IQ to be for the whole sample.

Tversky and Kahneman stated: "The correct answer is 101. A surprisingly large number of people believe that the expected IQ for the sample is still 100" in Psychological Bulletin, vol. 76, 105--110 (1971). (http://pirate.shu.edu/~hovancjo/exp_read/tversky.htm)

Can anyone justify the answer of IQ 101? Is it possible to solve this problem without being given the standard deviation of the population?

Answer 1

We imagine, of course, that the mean of the other $49$ is still $100$. That is, we expect the total IQ of the sample to be $$49\times 100+150=5050$$ Thus we expect the sample mean to be $$\frac {5050}{50}=101$$

For intuition: Suppose your sample had only two people, $A,B$. We measure $A$ and get $150$. If you insisted that the mean of the sample had to be $100$ no matter what, then you'd suddenly conclude that $B$ must be $50$. But this is absurd. $A,B$ are independent from each other....measuring $A$ has no impact on $B's$ score. After measuring $A$ we still expect $B$ to score $100$ so, given $A's$ score, we now expect the sample mean to be $125$.

Answer 2

You expect the mean IQ of the remaining $49$ children to be $100$. Imagine that the first child gives one of his IQ points to each of the remaining $49$ children. Then we expect the remaining $49$ children have a mean IQ of $101$, and the first child now has an IQ of $101$, so all $50$ children together have a mean IQ of $101$.

Answer 3

The answer given in the book seems to assume the mean of the remaining students' IQ is 100 (more on that in a moment). Based on this assumption, one can calculate the mean for the entire 50 student sample by combining the IQ of the first sample mixed with the average IQ of other 49 students sampled as follows:

(150 + (49 × 100)) / 50 = 101

So according to the above calculation, the average IQ for the specific 50 student sample described in the book is 101. But I don't think this is correct because the answer in the book does not use all the information from the problem statement.

Specifically, the result from the book does not account for the fact that having identified the first sample as having a 150 IQ, we now know that the remaining population's IQ has now dropped to ever so slightly below 100.

This is because the overall population is given as 8th graders in a city. We don't know the city, but we can put some realistic boundaries around it. Let's say this can range from 50 to 1,000,000. I chose 50 as minimum population since otherwise our sample size is not possible, invalidating the conditions of the problem. And 1,000,000 is probably too high as an upper bound, but not absurdly high, given that the largest city in the world is Tokyo with a population of more than 33 million (according to "the internet").

With this consideration in mind, more precise answers can be calculated as:

(150 IQ + (49 * average IQ of remaining population)) / 50

where the remaining population's IQ is everyone's IQ except the first sample of IQ 150.

This remaining population's IQ is calculated as (recall mean IQ is 100, first sampled IQ is 150):

((TOTAL POP * 100) - 150)/(TOTAL POP - 1)

So for example, the remaining population's IQ when there are 50 students in the total population is:

((50*100) - 150)/(50-1) = 98.97959184

And when there are 100 students:

((100*100) - 150)/(100-1) = 99.49494949

Now we can combine all these insights to calculate the mean IQ of the 50 student sample and see that the sample mean varies based on total population size (total number of 8th graders that we sample from):

• population of 50: 100.0
• population of 60: 100.1694915
• population of 70: 100.2898551
• population of 80: 100.3797468
• population of 90: 100.4494382
• population of 99: 100.5
• population of 100: 100.5050505
• population of 1,000: 100.9509510
• population of 10,000: 100.9950995
• population of 100,000: 100.9995100
• population of 1,000,000: 100.9999951

So you can see as the population size goes to infinity the value will approach 101 (the answer from the book), but for cities with fewer than 99 8th graders, the value rounds down to 100.