1. Chapter 11 Inductive Reasoning Causal Reasoning
Part 2
Causal Reasoning
Calculating Statistical Probabilities
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Causal Statements
State the cause of something
Examples:
The floor is wet because the toilet is leaking.
I itch because I got a mosquito bite.
NOT causal statements!
The toilet is leaking because the floor is wet.
I got a mosquito bite because I itch.
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3. Forming Causal Hypotheses
A causal HYPOTHESIS is a causal statement offered for further investigation or testing.
These statements could be offered as hypotheses, rather than as claims.
The floor is wet because the toilet is leaking.
I itch because I got a mosquito bite.
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4. “Paired unusual events” principle
If something unusual happens, look for something else unusual that happens. It may be the cause.
Example:
As soon as my throat got scratchy, I took Zicam. My cold went away faster than usual.
:. Perhaps Zicam caused it to go away faster than usual.
There are three principles that suggest causal hypotheses. In 10th edition this one was called “Method of Difference”
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5. 2. “Common variable principle”
A variable common to multiple occurrences of something may be related to it causally.
Example:
When several people in Kearny complained to their doctors about acute intestinal distress, health officials investigated and found that all had eaten tacos at the county fair.
:. Perhaps the tacos caused the infection.
In 10th edition called “Method of Agreement”
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6. 3. “Co-variation principle”
If a variation in one phenomenon is accompanied by a variation in another, the two phenomena may be related causally.
Example:
Over the past few years online instruction has increased at Memphis State. GPAs have also gone up.
:. Perhaps the increase in the GPA was caused by the increase in online instruction.
In 10th edition also brought in under Method of Agreement
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7. Confirming Causal Hypotheses
To confirm a causal hypothesis you try to show that, but for the suspected cause, the effect of interest would not have happened.
This is the basic thing you want to show, to confirm a hypothesis. That the hypothesized cause is the conditio sine qua non.
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8. Randomized controlled experiment
An experiment in which subjects are randomly assigned either to an “experimental group” (E) or a “control” (C), which theoretically differ from one another in only one respect: subjects in the E group are subjected to a suspected cause.
Example:
In an experiment, 50 willing volunteers were infected with a cold virus and then randomly divided into two groups. The subjects in one group were given Zicam, as per instructions on the box. Two weeks later, the number of people with colds in both groups was compared. Eighteen of the subjects who had not been treated with Zicam had colds, and only 10 of the subjects who had been treated had colds, a difference that is statistically significant.
:. Zicam probably reduced the frequency of colds in the experimental group.
There are three principles (methods, arguments) used to Help confirm causal hypotheses. In 10th edition this was called “Controlled cause-to-effect experiment.”
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9. 2. Prospective observational study
A study of two groups, in one of which a suspected causal agent is universally present, and in the other of which it is universally absent. A significant difference in the frequency of a suspected “effect” in the comparison groups is evidence of causal linkage between it and the suspected causal agent.
Example:
Does partying on the weekends adversely affect academic performance? The academic performance of 100 students who identified themselves as attending parties on most weekends was compared with that of 100 students who said they rarely or never attended weekend parties. It was found that 60 percent of the party-goers had GPAs below the mean for all San Diego State University students. Only 30 percent of the students who said they rarely or never attended weekend parties had GPAs below the mean.
:. Attending weekend parties probably adversely affects a student’s academic performance.
In 10th edition this was called “Non-experimental Cause-to-effect study”
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10. 3. Retrospective observational study
A study of two groups, in one of which a phenomenon of interest is universally present, and in the other of which it is universally absent. A significant difference in the frequency of a suspected causal agent in the comparison groups is evidence of causal linkage between the it and the phenomenon.
Example:
Does partying on the weekends adversely affect academic performance? Students at San Diego State University were surveyed about their weekend study habits. It was found that of 100 students on academic probation, 60 percent identified themselves as attending parties on most weekends. By contrast, only 20 percent of 100 students not on academic probation identified themselves as attending parties on most weekends.
:. attending weekend parties probably adversely affects a student’s academic performance.
In 10th edition this was called “Non-experimental Effect-to-cause study.”
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11. Calculating Statistical Probabilities
We often have to make decisions based on probabilities. Examples:
The weather channel says there is an 80% chance of rain tomorrow. Therefore we should postpone the picnic.
Don’t bet on the 6; the 7 is more likely to come up than a 6.
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Independent events
Two or more events are independent when the outcome of one has no effect on the outcome of the other(s). Examples:
When a coin is flipped twice in succession, how the first flip comes up has no effect on how the second flip comes up.
Whether the Hot Springs Trojans win their game Friday night has no effect on whether the USC Trojans win their game Saturday.
To believe that one independent event’s outcome can influence another is to commit The Gambler’s Fallacy.
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13. Calculating probabilities of multiple independent events
Express the probability of each event as a decimal between 0 and 1, then
Multiply those probabilities to obtain the probability of all the events occurring.
Example:
Chance of flipping heads: .5
Chance of flipping heads a second time: .5
Chance of flipping heads twice in a row: .5 x .5
Chance of flipping heads three times in a row: .5 x .5 x .5
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14. Alternative occurrences
We may be interested in whether either of two alternatives will occur.
Example:
When we draw a card from a 52-card deck, what is the probability that it will be either a spade or a heart?
Since one-fourth of the deck is spades and one-fourth hearts, we simply add the probabilities together:
Chances that it will be a spade: .25
Chances that it will be a heart: .25
Chances that it will be either a spade or a heart: = .5
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Expectation Value
Expectation value (EV) is the amount one stands to gain combined with the chances of gaining it.
Example (the “Big 6” bet):
You stand to gain $5 if the dice come up 6 before a 7 and you stand to lose $5 if the dice come up 7 before a 6.
To find the EV, multiply the amount of gain by the likelihood of that gain, and subtract from that the result of multiplying the amount you might lose by the likelihood of the loss.
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16. Expectation Value (continued)
The probability of a six occurring before a seven is 5 to 6, or about (Typically, five of each 11 chances will come up 6; six of them will come up 7.)The chances of a seven occurring before a six is 6 to 5, or about
.55
If you win: 5 x .45 = 2.25
If you lose: 5 x .55 = 2.75
Subtract the lose number from the win number and get an EV of minus .5.
Any number greater than 0 is a good bet; any number less than 0 is a bad bet. The bet in question is a bad bet indeed. (To be a fair bet, you would need to win $6 if your 6 arrived before a 7 and lose only $5 if the 7 came first.)
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17. Calculating Conditional Probabilities
Given that a certain percentage of Xs are Ys, how can you calculate the percentage of Ys that are Xs? To arrive at this answer we must calculate a conditional probability.
The key: convert the question to one about proportions.
We already know: the percentage of Xs that are Ys.
We need to know:
How many Xs are there per 1,000 population
The percentage of non-Xs that are Ys
(If we try to answer the question without knowing item 1, we commit the fallacy of overlooking prior probabilities, and if we do so without knowing item 2 we commit the fallacy of false positives—see Chapter 8.)
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18. Calculating Conditional Probabilities, continued
Example: We learn that 18% of people who had flu shots in your town got the flu. What is the probability that someone who got the flu had a flu shot?
To relate this to the previous slide, Xs are people who had flu shots, and Ys are people who got the flu.
So we need to know:
How many people per 1,000 had flu shots
How many people who did not have flu shots got the flu
From this information we can calculate the proportion of people who got the shots who also got the flu. See the next slide.
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19. Calculating Conditional Probabilities, continued
Say we learn that 100 out of every 1,000 people had flu shots. 1
We knew that 18% of them got the flu. That’s 18.
Let’s also say we learn that 24% of the 900 people who did not have flu shots got the flu. That’s 216.
Now we calculate:
18% of the 100 who had flu shots got the flu: 18
24% of the 900 who did not have shots got the flu: 216
So, of the 234 people who got the flu, 18 got the shots, around 7%
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