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Summary SMCR - statistical modeling for communication research
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Chapter 1: Sampling distribution: how different could by sample have been?1.1 statistical inferences: making the most of your data
Inferential statistics It provides us techniques to make statements about a large set of
observations (population) from a smaller set of observations (sample).
1.2 A discrete random variable: How many yellow candies in my bag?
If we draw two random samples from the same population, we will likely obtain different samples.
This means that two random samples from a population do not need to be identical.
Random sample by chance selecting a part of the population to sample
1.2.1 Statistical inference: Making the most of your data
Sample statistic is a value describing a characteristic of a sample, it is also called a random variable
because the number varies depending on the sample and is random because the score depends on
chance.
The sampling space refers to all possible outcomes for the sample statistic. So, if a bag contains 10
candies, and we want to know how many candies are yellow, the sampling space includes 1, 2,3, 4,
5, 6, 7,8, 9,10-the bag can contain these numbers of yellow candy.
1.2.2 Sampling Distribution
If we draw many random samples from a population and put the frequencies of the sample statistics
into a chart, we obtain a distribution of the outcome scores of many samples, called a sampling
distribution. The sampling distribution represents an infinite number of samples.
In a graph you find…
Sampling space for the sample statistic by looking at the horizontal axis.
The vertical axis shows the number of samples that have been drawn with a particular
sample statistic.
1.2.3 Probability and Probability Distribution
Sampling distributions contain absolute frequencies. If we change these frequencies into
proportions, we obtain a probability distribution of the sample statistic, which contains a sample
space with a probability between 0 and 1 for each outcome of the sample statistic.
How to tell if it is a probability distribution?
looking at the vertical axis: probabilities (between 0 and 1) instead of frequencies
If only a limited number of outcomes are possible we refer to the probability distribution as a
discrete probability distribution.
Continuous probability distribution A sampling space with a probability (between 0 and 1) for
each outcome of the sample statistic.
What does the probability distribution tell us?
The outcomes we can expect
The probability that a particular outcome may occur
1.2.4 Expected value or Expectation
If in the sampling distribution, the curve peaks at 2, we can expect to find two yellow candies in our
bag. In other words, we expect the proportion of yellow candy in the sample to equal the population
proportion.
, The mean of the sampling distribution equates to the expected
value of the sample statistic. The expected value refers to the mean of the
sampling distribution of a random variable. So, if the mean of the sampling
distribution is 2, we expect to obtain two yellow candies from our bag.
The mean of the sampling distribution is equal to the population proportion. The mean of the
sampling distribution (the expected value) = the population value. Only the case is the sample
statistic is an unbiased estimator of the population value.
1.2.5 Unbiased Estimator
Most sample statistics are considered unbiased estimators of the population statistic (parameter).
Keep in mind that we look at the proportion of yellow candy in our sample, not number: if we have
two yellow candies in our bag,we look at the proportion of yellow candy in the sample-in this case,
2/10, so .2-and refer to this proportion as an unbiased estimator of the population proportion.
1.2.6 Representative Sample
The sample is usually not completely representative since it is random and depends on chance. We
can expect the random sample to be representative of the population, so we say that the sample is
in principle representative of the population.
1.3.1 Continuous Variable
Weight is a continuous variable, meaning the variable does not have absolute frequencies.
1.3.2 Continuous sample statistic
For a continuous sample statistic, many things are the same as they were for a discrete sample
statistic, such as candy colour: average weight of all candy in the population = to the average weight
of the population, which is the average weight we expect in our sample (the expected value).
1.3.3 probability density
Probability density a means of getting the probability that a continuous random variable (like a
sample statistic) falls within a particular range
If there is a label to the vertical axis of a continuous probability distribution it is probability
density
But what if we want to know what the probability of obtaining a sample with a particular average
candy weight?
Look at a range of values. Average weight of 2.8 gram, we can set a different standard that examines
a range of values: the probability of obtaining a bag that has at least 2.8 gram or at most 2.8 gram.
1.3.4 p values
To display this type of probability in a sampling distribution, we have to look at the area between the
horizontal axis and the curve. The area beneath this curve is called the probability density function.
The area beneath a curve is always equal to 1, meaning the area belonging to a set of values will be
equal to 1 or less.
If the probability concerns the right-tail of the sampling distribution, meaning it specifies a threshold
value (2.8 and all greater values), it is a right-hand probability. If it
concerns the left- tail-2.8 and all lesser values-it is a left-hand
probability.
,The probability of values that include the threshold values, values under and above it are called p
values. The probability of values up to the threshold value is referred to as left-hand p value, and
the probability of values above the threshold is referred to as right-handed p value. The area under
a probability distribution curve will always add up to 1.
1.3.5 Probabilities always sum to 1
1.4 Concluding remarks
In a sampling distribution, samples are our units of analyses, and sample characteristics are our
observations. Each sample is one observation of the sampling distribution. The average of sampling
distribution is the mean of means. The sampling distribution links the sample to the population.
Chapter 2: Probability models: how do I get a sampling distribution
How should we create a sampling distribution based on a single sample?
1. Bootstrapping
2. Exact approaches
3. Theoretical approximations.
2.1 The bootstrap approximation of the sampling distribution
In bootstrapping, we draw a large number of samples from the first sample.
For these sub-samples, we should find out the sample statistics and use these in our
sampling distribution.
A bootstrap sample should be as large as the primary sample. The bootstrap sampling distribution
does not need to look like the true sampling distribution, since it is not likely it will exactly mirror the
population.
2.1.1 Sampling with and without replacement
All bootstrapped samples must be as large as the original sample. If we sample without
replacement, new samples will be identical to the initial sample. If we sample with replacement,
this means we can draw an observation more than once, so the new sample will look different than
the original sample.
We often use sampling without replacement; however, statistically, we use sampling with
replacement so our bootstrap samples can look different from our original sample.
2.1.2 Calculating probabilities with replacement
We usually calculate probabilities with the assumption that we sampled with replacement.
2.1.3 Calculating probabilities without replacement
In research, we sample without replacement, but we for statistics, we sample with replacement. This
is fine if the population is much larger than the sample
2.1.4 Limitations to bootstrapping
For the bootstrapped sampling distribution to resemble the true sampling distribution, we must
draw large samples.
If our sample is large, the bootstrapped sampling distribution will look similar to the true sampling
distribution.
If our sample is small, it likely will not.
, For them to look alike, the initial sample must be nearly representative of the population.
Never know whether our sample is representative of the population, this is the biggest
limitation of the bootstrap approach.
2.1.5 Any sample statistic can be bootstrapped
We can retrieve a sampling distribution for any sample statistic of interest. Bootstrapping is the only
method we have to retrieve a sampling distribution for the sample median.
2.2 Bootstrapping in SPSS
How to bootstrap in SPSS (completing an independent samples t-test with bootstrapping):
1. Analyse>Compare Means>>Independent-Samples T Test
2. Dependent variable in Test Variable box and put independent variable in Grouping Variables
box
3. Define groups>>select Bootstrapping>>check Perform bootstrapping>>usualily 5000 is the
number of samples we want>>click on Bias corrected accelerate
4. Click on continued
5. Paste
6. Run
Interpreting results:
Levene's test on homogeneity of variances is not statistically significant >> may assume that the
population variances of the two groups are equal>> interpret top row in bootstrap table.
Check mean differences >> if very little value, very small difference
Check confidence interval in bootstrap table >> to y >> with 95% confidence we can say that __ are
on average x ___ or up to ___. We cannot tell which groups is [heavier] in the population with
sufficient confidence.
Median:
[Check values then bootstrap]>> descriptive >> frequencies > select median >> bootstrap data for
missing analyze descriptive frequencies >>
Interpreting Results
Median__ in the sample is __. With 95% confidence we expect median ___ to be between ___ and
__ in the population.
2.3 Exact approaches to the sampling distribution
If we are sure of the sample statistic in the population, we can exactly calculate the probability in the
sample.
2.3.1 Exact approaches for categorical data
This approach lists all possible combinations, which can only be down with categorical or discrete
variables
2.3.2 Computer-intensive
We can use the exact approach on discrete and categorical variables because they have a limited
number of values. If there are many categories, this approach can take a lot of computing power.
2.4 Exact approaches in SPSS
How to compete an exact approach in SPSS (completed while doing a chi-square test):
1. Analyse>>Nonparametric Tests>>Legacy Dialogs>>Chi-square
2. Click on Exact
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