David W. answered • 03/16/20
Experienced Prof
"If students can major either in Humanities or Sciences, which fact establishes that the observation 'a student is male' is independent from the observation 'a student is a Science major.'
Please be very careful using the word "establishes." [note: statisticians often do not]
First, there is the population, In this problem, "a student" is a sample of the population of students being considered. You might also have "selected students" from the student population. When you consider the logical argument, "If A then B," you must consider every student in the population. For large populations or for inconvenient situations, we take a sample and use statistical inference (that is, probability).
Remember the story of the ignorant farmboy who said, "All ripe apples are red, so I won't eat the green apple you are offering me because it is not ripe."
If there is a random number generator (on a computer, it is called a pseudo-random number generator because it only approximates randomness), it is very unlikely to generate 50 numbers that are all even. Rather, I would expect, statistically (by theory and by observation) that half of the numbers generated will be even and half odd because that situation happens most often. However, generating 50 even numbers is possible. Also, generating 1000 even numbers is possible (but much less likely). Therefore, I cannot say that a generated random number is even. [note: However, I might observe that it is even for 50 or 1000 times (you know, like your chances of winning the lottery.)]
One fact is that students may be divided into two (perhaps unequal) groups -- those that take Humanities and those who take Sciences. There is no check whether they are male or female when they sign up, but there may be another relationship that influences this decision (e.g., Humanities costs equal for everyone, but Sciences costs more for males because of special instruments/equipment).
Another fact is that students are identified as either male for female. This identification is done without asking which major they are taking.
Therefore, we would expect that these situations would happen most often (most likely; not 'establishes').
- The percentages of males and females in the general population produces that same percentage in all samples (like to determine which is their major),
- The percentage of students is each major produces that same percentage in all samples (like to determine whether a student is male or female).
So,
A. "As many males" says nothing about how many females (but percent does).
B, "As many males as females" says nothing about how many majors (but percent does).
*> C. "Males constitute the same fraction of students majoring in Humanities as in Sciences." means that it is just as likely that a selected male student majors in Humanities as majors in Sciences ('likely,' but not 'established.')
D. "The fraction of males among the students majoring in Humanities is equal to the fraction of females among the students majoring in Sciences."
Note that D would have been true if it had said, "The fraction of males among the students majoring in Humanities is equal to the fraction of males among the students majoring in Sciences."
