Psychology 302

Probability and Decision Making

Most of the statistics that we have talked about so far involve describing distributions of samples

or describing relationships within samples. These are examples of descriptive statistics. But

most of the time, we are actually interested in using the data from our sample to make an

inference to a population. In that case, we would be doing inferential statistics.

Inferential statistics are based on the principles of probability. In order to use a sample to make

an inference about a population, we need to consider the probability of different events occurring

in the population based on what we observe in our sample. Before we can do that, we should

start with a (very) brief review of some key terms in probability

The probability of an event occurring [p(A)] is equal to the relative frequency of the event in the

long run. For example, pass completion average = # passes completed divided by # passes

attempted. In the 2019 season, Russell Wilson attempted 516 passes, 341 of which were caught.

Thus, we predict that he has a 341/516 = .66 probability of completing the next pass he throws.

The limits of probability are 0 to 1. Probability of an event occurring plus probability of an

event not occurring equals 1.0. P(A) + P(not-A) = 1.0.

The rules of probability only apply to random events. Remember that a random sample

(sometimes called a “probability sample”) requires that all elements or individuals within the

population have an equal probability of being selected for the sample.

Probability Distributions

Empirical probability distribution = measured probability. This is a distribution based on

observation of actual events. Pass completion average is an example of this. Another example is

in Consumer Reports magazine where they report repair rates of various automobiles. Car

models with lower repair rates are expected to be less likely to need repairs in the future.

Theoretical probability distribution = based on theory. This is a distribution based on

assumptions about the probability of events occurring. It is NOT based on guessing! Theoretical

probability distributions can be created for events for which we have very accurate knowledge

about the probability of certain events occurring. For example, you know that a fair coin has .5

probability of coming up heads. You don’t need to toss a coin a thousand times to figure this

out, you just compute it by this formula:

# of outcomes that satisfy the event P(event) =

# of possible outcomes

So for the coin example, 1

.5 2

heads

heads tails  



P(rolling a six on a 6-sided die) = 1

.17 6 

Independent events are when the occurrence of one event does not influence the probability of

another event. Examples of this are coin tosses, dice rolls, and slot machines. (Mistaken beliefs

about “hot dice” or someone being “due” for a jackpot on a machine that hasn’t paid out in a

while are referred to as the Gambler’s Fallacy.)

Dependent events are when the occurrence of one event does influence the probability of another

event. Card games like Blackjack and poker are based on dependent events, because once

certain cards have been dealt, they cannot be dealt again in that cycle.

Sampling with replacement is when the selected sample is returned to the population before the

next sample is drawn.

Sampling without replacement is when the selected sample is not returned to the population

before subsequent samples are drawn. The probability of events occurring changes with the new

samples. For example, say you have a raffle with three prizes. First prize winner is not eligible

for the other two prizes, second winner is not eligible for the third prize.

If they sell 100 tickets:

P(1st prize) = 1/100

P(2nd prize) = 1/99

P(3rd prize) = 1/98

Probability and the Standard Normal Curve

The standard normal curve is a theoretical probability distribution. It specifies the theoretical

probability of having certain values within a distribution. We can use the normal curve to

determine probabilities associated with events that are approximately normally distributed.

Say IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. (this

is pretty much true.)

What’s the probability of drawing one person at random from the population who has an IQ of at

least 110 (or higher)? To answer this, we need to compute a z-score for that person:

110 100 .67

15

X

X

X Z





    

Looking in the Z-table, we see that the probability of having a Z-score of .67 or greater (area in

the tail above) is p = .2514. So there is about a 25% chance of randomly grabbing a person with

an IQ of 110 or greater from the population with a mean of 100 and standard deviation of 15.

What we just did refers to determining probability of single observations. But often we want to

know probabilities associated with means.

What’s the probability of drawing a random sample of 16 people who have a mean IQ of at least

110? You might guess that this probability will be smaller than the probability of getting just

one person with an IQ of at least 110. Of course, some of these 16 people could have IQs of less

than 110, but then some would need to have IQs of greater than 110 to balance it out, so that the

group mean is at least 110.

In order to answer this question, you need to remember what we talked about back before Test 1,

sampling distributions.

A sampling distribution of the mean is a theoretical distribution. It is based on what the

distribution of means would look like if you took an infinite number of samples of size N from

the population. We never actually bother to do that (who has time?) but we know that if we did,

in theory, the distribution would have some specific properties.

Remember that a sampling distribution has a Mean and variability. The Mean of the sampling

distribution is equal to . The variability of the sampling distribution is smaller than the

variability of the population.

The variability of the sampling distribution is called the Standard Error. In this case, because we

are calculating estimates of the Mean, the variability is the Standard Error of the Mean.

Note that this is different than the Standard Deviation of the sample or the population. (It is also

different than the Standard error of the estimate that we learned about with regression. I wish

these terms didn’t all sound so similar. But the difference between them all is very important, so

pay attention and be careful!).

Standard Error of the Mean is the variability of the sampling distribution of Means. It is the

standard deviation of the sampling distribution of means.

Because of the Central Limit Theorem, we know the following things:

1) The mean of the sampling distribution of means will be . Any given sample mean will

not be exactly the same as , but in the long run, they will average out to be exactly 

2) If you took an infinite number of samples of size N, the standard error of the mean (i.e., the standard Deviation of the sampling distribution of means) would be:

X X

N

  

What that means is that the variability of the sampling distribution is smaller when your

samples are bigger. Bigger samples mean you are more likely to get a good (accurate)

estimate of the true population Mean.

3) As the size of the sample increases, the shape of the sampling distribution of the mean will approach normal. What’s really amazing is that this is true even if the shape of the

original distribution is not normal.

Because of the central limit theorem, we can use a single sample of size N to estimate properties

of the sampling distribution (rather than actually needing to take an infinite number of samples).

OK, so now that you know about the sampling distribution of means, we can get back to our

earlier question about how to determine the probability of getting a sample of N = 16 people who

have an average IQ of at least 110.

We’re going to create a Z-score like we did before, except that now we will create a Z-score

based on a sampling distribution of means with  = 100,  = 15 and N = 10. Instead of

comparing one person’s score (X) to the sample mean, we will be comparing the sample mean to

the mean of the population. Instead of dividing by the standard deviation of X, we will divide by

the standard error of the estimate. The general formula for the Z-score will be:

X

X

X Z





  which is equivalent to X

X

X

N







For our example, 110 100 10

2.67 15 3.75

16

Z 

  

What’s the probability of getting a Z-score of 2.67 or greater? Look in the table, p = .0038. It is

a LOT less likely that we would get a sample of 16 people with a mean IQ of 110 than it is that

we would get one single person with an IQ of 110 (p = .2514).

Deciding whether a sample represents a population

So, if the probability of getting 16 people with a mean IQ of at least 110 just by chance is only

.0038, that might make you start thinking that maybe there is something other than just chance

operating here. Maybe those 16 people weren’t actually randomly sampled from a population

with a mean of 100 and a standard deviation of 15. Maybe those 16 people do not represent the

general population of people in the U.S. Maybe they actually represent some other population,

such a population of college students who have higher than average IQs.

We get suspicious about the representativeness of the sample because the probability of

obtaining a sample with those characteristics (mean IQ of 110) is very unlikely if those people

were really just randomly sampled from the general population with  = 100 and  = 15. But

what exactly do we mean by very unlikely? How unlikely does an event have to be before we

start getting suspicious?

Suppose that the sample of 16 people we drew only had a mean IQ of 101. Would we be

suspicious that they were not really representative of the general population? Let’s see:

101 100 1

.27 15 3.75

16

Z 

   the probability of getting a Z of .27 or greater is .3936.

If there’s a 39% chance that we could have found 16 people with a mean IQ of 101, it doesn’t

seem so strange that it could have happened just by chance.

So what probability do we want to use as our cutoff for “too unlikely?” Well, the conventional

level in social sciences is usually p = .05 or less (that is, p < .05).

Often, it is easier to think about this in the opposite way, by asking what is the Z-score that will

give us exactly 5% probability in the tails of the distribution? This is referred to as the critical

value.

One important question is whether we just want our 5% to be in one tail of the distribution or

whether we want it to be divided up between the top and bottom end of the distribution. In this

case, we probably would have been just as surprised if we randomly sampled a group of 16

people and found that they had an average IQ of 90 (10 points below the mean instead of 10

points above the mean). So we would think things are very unlikely if they happened to be much

higher than average as well as much lower than average. So we will split our 5% up between the

two tails of the distribution.

If we have 5% split up between the two ends of the distribution, that means we have 2.5% or

.025 in each tail of the distribution. Let’s find the critical value. What is the Z-score that has

.025 in the tail beyond it? Look in the table, it’s Z = 1.96.

We will refer to the area beyond 1.96 (that is, less than –1.96 or greater than +1.96) as the

region of rejection. If we end up getting a sample that gives us a Z-score that is in the region of

rejection, we will conclude that it is too unlikely that this could have happened just by random

chance. There must be something else going on, such as the group we sampled not actually

being representative of the general population.

And this leads us right into the topic of hypothesis testing, which we’ll discuss next.

.025 .025

-1.96 +1.96