This is a summary of the book "A Gentle but Critical Introduction to Statistical Inference, Moderation, and Mediation", which is the course material for the course "Statistical Modelling for Communication Research" at the UvA.
* Summary includes all 11 chapters :)
Sampling Distribution: How Different Could My Sample Have Been?
(Chapter 1)
Saturday, 29. August 2020 1:40 PM
Statistical Inference: Making the Most of Your Data
• Statistics offers techniques for making statements about a larger set of
observations from data collected for a smaller set of observations
○ The large set of observations about which we want to make a statement
is called the population.
○ The smaller set is called a sample.
We want to generalize a statement about the sample to a statement
about the population from which the sample was drawn.
• Statistical inference: Generalization from the data collected in a random
sample to the population from which the sample was drawn
A Discrete Random Variable: How Many Yellow
Candies in My Bag?
• If we draw random samples from the same population, we are likely to obtain
different samples
Sample statistic
• We are usually interested in a particular characteristic of the sample rather
than in the exact nature of each observation within the sample
• Sample statistic: A number describing a characteristic of the sample
○ Sampling space: All possible outcome scores
• Random variable: The sample statistic
○ A sample statistic is a variable because different samples can have
different scores
A random variable because the score depends on chance, namely
the chance that a particular sample is drawn
•
Sampling distribution
• Some sample statistic outcomes occur more often than other outcomes
• Sampling distribution: The distribution of the outcome scores of very many
samples
Chapter 1 Page 1
, •
○ Numbers on the horizontal axis constitute the sampling space, that is, all
values that the sample statistic
○ Left-hand vertical axis shows the number of samples that have been
drawn with a particular value for the sample statistic
○ The right-hand vertical axis gives the proportion of previously drawn
samples with a particular number of x
The proportions can be interpreted as probabilities, namely the
probability that a previously drawn sample contains a particular
number of x
Probability and probability distribution
•
○ Number of yellow candies in the sample (bag) is the sample statistic
○ Set of all possible outcomes of the sample statistic is the sampling space
○ Example of a discrete probability distribution because only a limited
number of outcomes are possible
• The sampling distribution tells us all possible samples that we could have
drawn
• We can use the distribution of all samples to get the probability of buying a
Chapter 1 Page 2
, • We can use the distribution of all samples to get the probability of buying a
bag with exactly five yellow candies from the sampling distribution
○ We divide the number of samples with five yellow candies by the total
number of samples we have drawn
• If we change the (absolute) frequencies in the sampling distribution into
proportions (relative frequencies), we obtain the probability distribution of the
sample statistic
○ Probability distribution of a sample statistic: Sampling space with a
probability (between 0 and 1) for each outcome of the sample statistic
• Sampling distributions tend to have proportions, that is probabilities, instead
of frequencies on the vertical axis
• The sampling distribution as a probability distribution conveys very important
information
○ Tells us which outcomes we can expect
○ Tells us the probability that a particular outcome may occur
• We may refer to probabilities both as a proportion, that is, a number between 0
and 1, and as a percentage
Expected value or expectation
• The value that we are most likely to encounter in the sample that is to be
drawn, intuitively must be related to the distribution in the population from
which the sample is drawn
•
○ We expect that the proportion of yellow candies in the sample equals the
population proportion, which initially is .2
In a sample of ten candies, the expected number of yellow candies
is two
○ Two is the outcome with the highest probability in the sampling
distribution
○ The expected number of yellow candies in a sample bag of ten candies is
ten times the population proportion, so the expected number of candies
in the sample changes in accordance with changes in the population
proportion
○ The mean of the sampling distribution of the sample proportion is equal
to the population proportion
○ We are equally likely to draw a sample with less yellow candies than the
expected proportion as a sample with more yellow candies
Mean represents the “balance point” of a distribution
○ The expected value (mean of the sampling distribution) only equals the
population value if the sample statistic is an unbiased estimate of the
population value (parameter)
• The expected value equals the mean of the sampling distribution
Chapter 1 Page 3
, • The expected value equals the mean of the sampling distribution
○ Expected value: The average of the sampling distribution of a random
variable
○ The sampling distribution is an example of a probability distribution
More generally, the expected value is the average of a probability
distribution
□ The expected value is also called the expectation of a
probability distribution
Unbiased estimator
• A sample statistic is an unbiased estimator of the population statistic if the
expected value (mean of the sampling distribution) is equal to the population
statistic
○ We usually refer to the population statistic as a parameter
• If we were to estimate the number in the population (the parameter) from the
number in the sample we are going to vastly underestimate the number in the
population
→ This estimate is downward biased: It is too low
○ In contrast, the proportion in the sample is an unbiased estimator of the
population proportion
• Sometimes, we have to adjust the way in which we calculate a sample statistic
to get an unbiased estimator
○ We must calculate the standard deviation and variance in the sample in a
special way to obtain an unbiased estimate of the population standard
deviation and variance
Representative sample
• We expect a random sample to resemble the population from which it is drawn
• A sample is representative of a population if variables in the sample are
distributed in the same way as in the population
○ We know that a random sample is likely to differ from the population
due to chance, so the actual sample that we have drawn is usually not
representative of the population
▪ But we should expect it to be representative, so we say that it is in
principle representative of the population
○ We can use probability theory to account for the misrepresentation in the
actual sample that we draw
▪ This is what we do when we use statistical inference to construct
confidence intervals and test null hypotheses
A Continuous Random Variable: Overweight And
Underweight
Continuous variable
• Weight is a continuous variable because we can always think of a new weight
between two other weights
○ If we can always think of a new value in between two values, the
variable is continuous
Continuous sample statistic
• The sample mean is an unbiased estimator of the population mean, so the
average weight of x in the population is the average of the (average x weights
in the) sampling distribution
○ This is the average weight that we expect in a sample drawn from this
population (the expected value or expectation)
Continuous probabilities --> got wrong in FMTest
• When we turn to the probabilities of getting samples with a particular average
weight, we run into problems with a continuous sample statistic
○ The probability distribution of the sampling space, that is, of all possible
outcomes, is going to be very boring: just (nearly) zeros
○ It will take forever to list all possible outcomes within the sampling
space, because we have an infinite number of possible outcomes
Probability density --> got wrong in FMTest
• We can solve the problem above (continuous probabilities) by looking at a
range of values instead of a single value
We choose a threshold and determine the probability of values above or
Chapter 1 Page 4
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