Download The Milgram Experiment

The Milgram Experiment
Stanley Milgram, a Yale psychologist, carried out an experiment in 1963 to investigate the conflict between obedience to authority and personal conscience.
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The participants in the experiment played the role of “teacher”.
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The experimenter told the teacher that they should administer an electric shock to the “learner” every time he made a mistake on a task, increasing the level of the shock each time to a maximum of 450 volts, which was labeled “danger – severe shock”.
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The learner was actually an actor and the electric shocks were not real.
From www.simplypsychology.org/milgram.html
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65% of participants obeyed authority (the experimenter) and administered the shocks to the maximum level. 35% refused to continue after reaching a lower voltage level.
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Each participant in Milgram’s experiment can be considered a trial.
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There are two possible outcomes for each trial: o refuses to administer the shock (“success”)
o administers the shock (“failure”)
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The probability of a randomly choosing a participant who will have the outcome of success is p=0.35.
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When a trial has two possible outcomes, with a constant probability of “success”, the trial is called a Bernoulli trial.
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If we code the outcome for a Bernoulli trial with the numeric values 1 for success and 0 for failure, we get a Bernoulli random variable.
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The probability distribution for the Bernoulli random variable is:
Outcome
Probability
1
0.35
0
0.65
Outcome
Probability
1
0.35
0
0.65
Let X be a random variable with the probability distribution given in the above table.
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Is X a discrete or continuous random variable?
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What is the expected value (or “mean” or “expectation”) of X?
E(X)=
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What is the variance of X?
Var(X)=
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What is the standard deviation of X?
SD(X)=
Suppose that we randomly select four individuals (A, B, C, D) to participate in this experiment. We will count the number of people in this sample who refuse to administer the shock, and call it the random variable Y.
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Is Y a Bernoulli random variable?
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What is the probability that exactly 1 of the four participants will refuse the administer the shock?
P(Y=1)=
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What did you have to assume to calculate P(Y=1)?
Suppose that we randomly select four individuals (A, B, C, D) to participate in this experiment. We will count the number of people in this sample who refuse to administer the shock, and call it the random variable Y.
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Is Y a Bernoulli random variable?
Suppose that we randomly select four individuals (A, B, C, D) to participate in this experiment. We will count the number of people in this sample who refuse to administer the shock, and call it the random variable Y.
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What is the probability that exactly 1 of the four participants will refuse the administer the shock?
P(Y=1)=
Suppose that we randomly select four individuals (A, B, C, D) to participate in this experiment. We will count the number of people in this sample who refuse to administer the shock, and call it the random variable Y.
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What did you have to assume to calculate P(Y=1)?
Y has a Binomial probability distribution.
Properties of a Binomial distribution:
• A simple random experiment with two possible outcomes (a.k.a. a Bernoulli trial) is repeated n times (n is fixed).
• The n trials are independent.
• For each trial, there is a probability of “success”, p.
• A Binomial(n,p) random variable is the count of the number of successes in the n trials.
What is the probability distribution of the random variable Y, the number of participants from A, B, C, and D who refused to administer the shocks?
Is Y a discrete or continuous random variable?
Calculating Binomial probabilities:
If Y~Binomial(n,p)
If there are k successes, then there are n‐k failures. P(Y=k) = (the number of ways you can have k successes in n trials) * pk (1‐p)n‐k
Continuous random variables: can take on any value (real number) in an interval.
Probabilities for continuous random variables are derived as areas under a “density function”.
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The figure below shows a probability density function for a continuous random variable, X. What are the properties of the plot that are necessary for it to be a valid probability density function?
0.06
0.08
a) P(X=5)
0.02
0.04
b) P(2 ≤ X ≤ 6)
0.00
Density
0.10
Find:
0
2
4
6
X
8
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The words “mean”, “variance”, & “standard deviation” can be used to describe the properties of a probability distribution. This is related, but different than how we use these words to summarize data.
The most important continuous probability distribution for us:
Normal(μ,σ2) or N(μ,σ2) (a.k.a. the Gaussian distribution)
μ is the “centre” of the distribution. (Note: the textbook writes N(μ,σ) )
If a random variable, X, has a Normal distribution, that is X~N(μ,σ2):
• it can be any value from ‐∞ to +∞
• E(X)=μ
• Var(X)=σ2
For a Normal random variable:
The probability of being within one σ of μ is 68%
The probability of being within two σ of μ is 95%
The probability of being within three σ of μ is 99.7%
A standard Normal random variable, Z, has mean 0 and variance 1.
Z~N(0,1)
Properties of mean and variance of random variables:
What’s another way to write Var(aX‐Y+b) if X and Y are independent?
An important application of the properties of Expectation and Variance:
Suppose X~N(μ,σ2)
X/σ has variance 1
X‐μ has expectation 0
So (X‐μ)/σ has the standard Normal distribution
Using the Normal probability table (for the standard Normal distribution)
The table gives the probability of being below the value in the margin of the table.
It gives the area under the standard normal density curve to the left of the value in the margin of the table.
Solving problems with the Normal distribution:
Suppose X~N(10,4)
What is P(X=8)?
Normal probability calculations:
1. Draw a picture.
2. Standardize.
3. Get the corresponding values from the standard Normal table.
The serum cholesterol levels of 12‐ to 14‐year‐olds follow a normal distribution with mean 162 mg/dl and standard deviation 28 mg/dl. a)
What percentage of 12‐ to 14‐year‐olds have serum cholesterol values 171 or more?
b) What percentage of 12‐ to 14‐year‐olds have serum cholesterol values 194 or less?
c)
What percentage of 12‐ to 14‐year‐olds have serum cholesterol values between 105 and 138?
d) What is the probability that a randomly chosen 13‐year‐old has a serum cholesterol value between 105 and 138?
e) What is the 80th percentile of the distribution of serum cholesterol values for 12‐ to 14‐year‐olds?
f)
Suppose that a 12‐year‐old and a 13‐year‐old are chosen at random. What is the distribution of the difference between their serum cholesterol values?