Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

PREVIEW
In this chapter we extend the topic of probability to cover
larger samples; specifically, samples that have more than
one score. Fortunately, you already know the one basic
fact governing probability for samples: Samples tend to
be similar to the populations from which they are taken.
For example, if you take a sample from a population
that consists of 75% females and only 25% males, you
probably will get a sample that has more females than
males. Or, if you select a sample from a population for
which the average age is µ = 21 years, you probably
will get a sample with an average age around 21 years.
We are confident that you already know this basic fact
because research indicates that even 8-month-old infants
understand this basic law of sampling.
Xu and Garcia (2008) began one experiment by showing
8-month-old infants a large box filled with ping-pong
balls. The box was brought onto a puppet stage and the
front panel was opened to reveal the balls inside. The
box contained either mostly red with a few white balls or
mostly white with a few red balls. The experimenter alternated
between the two boxes until the infants had seen
both displays several times. After the infants were familiar
with the boxes, the researchers began a series of test
trials. On each trial, the box was brought on stage with the
front panel closed. The researcher reached in the box and,
one at a time, drew out a sample of five balls. The balls
were placed in a transparent container next to the box. On
half of the trials, the sample was rigged to have 1 red and
4 white balls. For the other half, the sample had 1 white
and 4 red balls. The researchers then removed the front
panel to reveal the contents of the box and recorded how
long the infants continued to look at the box.
The contents of the box were either consistent with the
sample, and therefore expected, or inconsistent with the
sample and therefore unexpected. An expected outcome,
for example, means that a sample with 4 red and 1 white
ball should come from a box with mostly red balls. This
same sample is unexpected from a box with mostly white
balls. The results showed that the infants stared consistently
longer at an unexpected outcome (M = 9.9 seconds)
than at an expected outcome (M = 7.5 seconds), indicating
that the infants considered the unexpected outcome surprising
and more interesting than the expected outcome.
Xu and Garcia’s results strongly suggest that even
8-month-old infants understand the basic principles that
identify which samples have high probability and which
have low probability. Nevertheless, whenever you are
picking ping pong balls from a box or recruiting people
to participate in a research study, it usually is possible to
obtain thousands or even millions of different samples
from the same population. Under these circumstances,
how can we determine the probability for obtaining any
specific sample?
In this chapter we introduce the distribution of sample
means, which allows us to find the exact probability
of obtaining a specific sample mean from a specific
population. This distribution describes the entire set
of all the possible sample means for any sized sample.
Because we can describe the entire set, we can find probabilities
associated with specific sample means. (Recall
from Chapter 6 that probabilities are equivalent to proportions
of the entire distribution.) Also, because the
distribution of sample means tends to be normal, it is
possible to find probabilities using z-scores and the unit
normal table. Although it is impossible to predict exactly
which sample will be obtained, the probabilities allow
researchers to determine which samples are likely and
which are very unlikely.
7.1 Samples, Populations, and the Distribution
of Sample Means
LEARNING OBJECTIVE
1. Define the distribution of sample means and describe the logically predictable
characteristics of the distribution.
The preceding two chapters presented the topics of z-scores and probability. Whenever a
score is selected from a population, you should be able to compute a z-score that describes
exactly where the score is located in the distribution. If the population is normal, you also
should be able to determine the probability value for obtaining any individual score. In a
normal distribution, for example, any score located in the tail of the distribution beyond
z = +2.00 is an extreme value, and a score this large has a probability of only p = 0.0228.
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