Chance News 68
"If you're a politician, admitting you're wrong is a weakness, but if you're an engineer, you essentially want to be wrong half the time. If you do experiments and you're always right, then you aren't getting enough information out of those experiments. You want your experiment to be like the flip of a coin: You have no idea if it is going to come up heads or tails. You want to not know what the results are going to be." Peter Norvig, as quoted at Slate Magazine. Submitted by Steve Simon.
A blogger responded to a John Allen Paulos article in The New York Times of October 24, 2010:
”One can't really say anything meaningful about probability without actual data.”
Submitted by Margaret Cibes
Paulos on probability
“Stories vs. Statistics”
by John Allen Paulos, The New York Times, October 24, 2010
In this article, Paulos discusses two classic probability problems, and 143 bloggersrespond en masse.
PROBLEM 1. With respect to the story of Linda the famous feminist bank teller, Paulos says that, while more details about a fictional character may make the story more believable, “the more details there are in a story, the less likely it is that the conjunction of all of them is true.”
In the classic illustration of the fallacy …, a woman named Linda is described. She is single, in her early 30s, outspoken, and exceedingly smart. A philosophy major in college, she has devoted herself to issues such as nuclear non-proliferation. So which of the following is more likely?
a.) Linda is a bank teller.
b.) Linda is a bank teller and is active in the feminist movement.
Paulos concludes, “Although most people choose b.), this option is less likely since two conditions must be met in order for it to be satisfied, whereas only one of them is required for option a.) to be satisfied.”
PROBLEM 1. Bloggers
No. 10. a.) is not necessarily more likely than b.). Paulos’ “understanding of others is weak,” and we need more information about how her activism may have led to sexual discrimination to answer the question.
No. 20’s response to #10. “Unless you believe that there could not possibly be such a person, Paulos is correct.”
PROBLEM 2, with variations. With respect to the classic two-boy problem, Paulos says that “our judgment of [a] probability is almost always affected by its intensional [sic] context.”
Given that a family has two children and that at least one of them is a boy, what is the probability that both children are boys? The most common solution notes that there are four equally likely possibilities — BB, BG, GB, GG …. Since we’re told that the family has at least one boy, the GG possibility is eliminated and only one of the remaining three equally likely possibilities is a family with two boys. Thus the probability of two boys in the family is 1/3. …. What if instead of being told that the family has at least one boy, we meet the parents who introduce us to their son? Then there are only two equally like possibilities — the other child is a girl or the other child is a boy, and so the probability of two boys is 1/2.
Paulos poses a “new variant of the two-boy problem”:
A couple has two children and we’re told that at least one of them is a boy born on a Tuesday. What is the probability the couple has two boys? Believe it or not, the Tuesday is important, and the answer is 13/27. If we discover the Tuesday birth in slightly different intensional [sic] contexts, however, the answer could be 1/3 or 1/2.
PROBLEM 2, with variations. Bloggers
No. 5. “[B]irth order is not part of the question, only whether the unidentified child is male or female, in which case the odds are 1 in 2 ….”
No. 7. Settembrini gives the calculations for the 13/27 problem.
No. 25, Ivan of NYC. “There is another version to the BG problem: suppose that in a family of two children, one is a girl named Emily. What is the probability that the other child is her sister? Turns out to be 1/2. (I was asked this question on an interview)”
1. How would you respond to Blogger 10's comment about needing more information?
2. With respect to blogger 20, suppose that you believe that there could not possibly be such a person as Linda. What would be the probability of a.)? The probability of b.)? Would Paulos be correct in this case?
3. How would you respond to Blogger 5's comment about birth order?
4. Without viewing Blogger 7’s calculations, calculate the answer to the 13/27 problem. Do you agree with Blogger 7's calculations?
5. Do you agree with Blogger 25 about the probability when you know one child's name?
Submitted by Margaret Cibes
Student learning lags over summers
“The Case Against Summer Vacation”
by David Von Drehle, TIME, Thursday, July 22, 2010
This article describes research about the effect on learning of long summer vacations for U.S. school children, especially for, but not limited to, low-income children without access to activities such as camp, travel, or other enrichment experiences.
[W]hen American students are competing with children around the world, who are in many cases spending four weeks longer in school each year, larking through summer is a luxury we can't afford. …. By the time the bell rings on a new school year, the poorer kids have fallen weeks, if not months, behind. And even well-off American students may be falling behind their peers around the world. …. "We expect that athletes and musicians would see their performance suffer without practice. Well, the same is true of students, [says one educational administrator]."
The article provides two graphics, neither of which is online now. The first was a time-series graph showing the gap among low-, middle-, and high-income math test scores over grades 1 through 5; the contributor (email@example.com) has an electronic copy for interested readers. The second was a bar chart containing the following data. (No information was given about the instrument on which the math scores were based or the number of students who were tested in each country.)
Country-School Year (median days)-Total Instructional Hours-Math Scores (15-year-olds)
For each country, assume that similarly large numbers of students in each country took a common math test and that median days in a school year was a pretty accurate representation of the length of the school year across that country.
1. Given that the correlation between median days in a school year and total instructional hours is about -0.007, what might you conclude about the relationship between days and hours in a school year? Suggest an explanation for this.
2. Sketch a graph of median days in a school year as the explanatory variable and math scores as the response variable.
a. The data points for which two countries fall outside of the relatively linear pattern of the other twelve countries?
b. Without those two data points, the correlation increases from about +0.108 to about +0.919. Can you think of any legitimate reason to omit them in order to report (with an appropriate note) a stronger correlation? What else might you want to know in order to decide?
c. The author suggests that spending 4 weeks longer in school might improve U.S. math scores. Do you see any evidence here that extending the U.S. school year by 4 weeks (20 school days) would improve U.S. math scores?
3. Given that the correlation between total instructional hours and math scores is about -0.397, what does this number alone imply about the relationship between hours in a school year and math scores? Do you see any evidence here that increasing hours would improve U.S. math scores?
4. For each country, a blogger calculated the total instructional hours of teaching required to increase a math score by 1 point and concluded, “The US is more efficient in its teaching then Mexico and Brazil.”
a. The U.S. has more instructional hours per point than most other countries shown. Do you agree that this higher ratio indicates more "efficiency"?
b. Do you agree with the blogger’s arithmetic with respect to his ordering of the U.S., Mexico, and Brazil ratios?
c. What, if anything, might the U.S. rank with respect to these ratios suggest about U.S. teachers and/or students, compared to those of other countries?
5. What other information and/or data might be helpful to decision-makers considering a decision about increasing the length of the U.S. school year?
Submitted by Margaret Cibes
Facebook data on relationship breakups
David McCandless has a hobby that many would find odd.
David McCandless, a London-based author, writer and designer, is constantly playing with data sets available online and translating heaps of code into well-designed visual stories. Some of Mr. McCandless’s notable projects include visualizing the billions of dollars spent by people and governments around the world and visually explaining the different views of United States politicians, divided by their political predilection.
Facebook, a social network site, has data on social status: single, in a relationship, married, it's complicated, etc. With the help of Lee Byron of Facebook, he produced the following graph on breakups by looking at changes in social status.
The repeated peaks are Mondays, a day at higher risk of breakups. The general findings are that
most breakups occur three times in the year — in the weeks leading up to spring breaks, right before the start of the summer holidays and a couple of weeks before Christmas.
1. Changing one's status in Facebook is a surrogate measure of the actual ending of a relationship. What limitations does this produce in this data set?
2. Mr. McCandless claims that "people tend to break up with their significant others on Mondays, presumably after a weekend grapple." Is there an alternate explanation for the spike of breakups on Monday?