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Test-preparation organizations like Kaplan, Princeton Review, etc. often advertise their services by claiming that students gain an average of 100 or more points on the Scholastic Achievement Test (SAT). Do you think that taking one of those classes would give a test taker 100 extra points? Why might an average of 100 points be a biased estimate?
A few random results for you to ponder…do the results justify the conclusions? Why or why not? Pick one or two statements and give a reason why the conclusions don’t necessarily follow from the statement
1. In the NFL, teams win more often when they score 13 points than when they score 14. Thus, scoring points is bad.
2. Often when people use regression analysis to estimate the effect of police officers or police spending on crime, they find that cities with larger police forces/budgets have higher crime rates. Therefore, police cause crime.
3. As ice cream sales increase, so to do drowning deaths. Thus, ice cream causes people to drown.
4. Studies find that students who drink more tend to have lower grades. Therefore, drinking leads to poor student performance.
5. Over the past 300 years, there has been a decrease in the number of pirates on the high seas, along with an increase in average global temperatures. Thus, the reduction in piracy has led to global warming.
6. Did you know there is a health benefit to winning an Oscar? Doctors at Harvard Medical School say that a study of actors and actresses shows that winners live, on average, for four years more than losers. And winning directors live longer than non-winners.
7. Children who come further down in the birth order have, on average, lower IQs than those born earlier in the birth order (e.g. First born children vs. 5th born children). Therefore birth order determines intelligence
Consider the following:
There is a game called Under-or-Over Seven. A pair of fair dice is rolled once and the resulting sum determines whether the player wins or loses his or her bet. For example, the player can bet $1 that the sum will be under 7 — that is 2, 3, 4, 5, or 6. For this bet, the player wins $1 if the result is under 7 and loses $1 if the outcome equals to or is greater than 7. Similarly, the player can bet $1 that the sum will be over 7 — that is 8, 9, 10, 11, or 12. Here the player wins $1 if the result is over 7, but loses $1 if the outcome is 7 or under. A third method of play is to bet $1 on the outcome 7. For this bet, the player wins $4 if the result of the roll is 7 and loses $1 otherwise.
a) Construct the probability distribution representing the different outcomes that are possible for a $1 bet on under 7.
b) Construct the probability distribution representing the different outcomes that are possible for a $1 bet on over 7.
c) Construct the probability distribution representing the different outcomes that are possible for a $1 bet on 7.
d) Show that the expected long run profit (or loss) to the player is the same, no matter which method of play is used.