CMNS 260 Lecture Notes - Lecture 11: Descriptive Statistics, Statistical Hypothesis Testing, Null Hypothesis
8 May 2018
School
Department
Course
Professor
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Quantitative Data Analysis (Multivariate Analysis and Introduction to Inferential Statistics)
Thinking about relationships by analyzing patterns in statistical tables of two and three variables
• Example of data from old class quiz about grades (cumulative GPA) and time spent on social
media
• Possible research question: Does the amount of time spent on social media have a relationship
with grades?
• Potential hypothesis: Students who spend more time on social media may spend less time
studying and have lower grade point averages.
Adding a third variable (place of birth)
• But what if this varies by place of birth (with people who were born elsewhere spending more
time on social media)?
When we take more variables into account it may be difficult to track relationships
• Some issues:
o How to track and follow patterns of many variables?
o What if there are relationships among the variables?
o What if the introduction of new variables changes the relationships?
• Many methods for trying to describe and analyze relationships among many variables.
From Bivariate to Multivariate Statistics (statistics with more than 2 variables)
• Analysis of relationships between two variables is a step towards studying relationships among
more than two variables
• Early efforts to produce graphic methods of analysis were often complicated
• Other statistical techniques developed to calculate trends, risks and probabilities
• One type of calculation is the marginal prediction rule
o Calulatios use oeall pattes of distiutio i the agials as asis for predicting
expected patterns in data collected
Estimating patterns when there is no relationship using expected counts: example of case with no
variation
• The Cout if this ee sue data, this ould e ho a espodets atuall aseed in
a certain way) in each cell adds up to the Total Count.
o Expected count: (row total) (column total) / N
• Row percentages add up to 100% across the row
o If the rows total 100 percent each, it has been percentaged across.
• The rule, then, is as follows:
1. If the table is percentaged down, read across.
2. If the table is percentaged across, read down.
• When analyzing the table, remember to compare row percentages up and down within a
column
Other Methods: Proportional Reduction of Error (PRE)
• Some Measures of assoiatio use idea of popotioal edutio of eo P‘E
• Proportional reduction of error statistics based on different approaches to:
1. Marginal Prediction Rule: Estimation of overall distribution of dependent variable with
observed variations in bivariate crosstabulations (e.g. using averages)
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2
2. Relational prediction rules (based on how independent and dependent variables are
connected)
3. Coutig eos i peditio usig oth the agial ad elatioal peditios
4. Standard Calculation used:
▪ E1 = errors using only dependent variable information
▪ E2 = errors using dependent & independent variable information
x² (Chi-square) tests
• In descriptive statistics a Chi-square statistical test is commonly used to compare observed data
with data we would expect to obtain according to a specific hypothesis.
• In inferential statistics the chi-square test is always testing what scientists call the null
hypothesis, which states that there is no significant difference between the expected and
observed result.
• (Note: There are ways to make more detailed calculations to take into account issues related to
saplig that ot e oeed i ou ouse.
Chi-square test for independence
• The chi-square test of independence statistic is computed by summing the difference between
the expected and observed frequencies for each cell in the table divided by the expected
frequencies for the cell.
• The test of independence assesses whether an association exists between the two variables by
comparing the observed pattern of responses in the cells to the pattern that would be expected
if the variables were truly independent of each other.
• Most statistical tests for independence follow a normal curve. If the statistical test shows that a
result falls outside the 95% region, you can be 95% certain that the result was not due to
random chance, and is a significant result
Logical foundations of tests of statistical significance: comparing expected results of null hypothesis to
other patterns
• Assumptions regarding independence of two variables
• Assumptions regarding representativeness of samples drawn using probability sampling
techniques
• Observed distribution of sample in terms of two variables
• Example: Study of a population of 256 people, half male & half female about comfort using
Internet to make purchases
Studying relationships between two variables by thinking about expected count (if no relationship)
• Uses of cross-tabulations and graphs when to examine patterns of relationships between two
variables (bivariate descriptive statistics).
• When change in one variable is associated with a systematic change in another variable the
statistical relationship is called a correlation.
• But using expected count can help you think through patterns
Using statistical tests to assess whether sample accurately represents trends in a population
• Are patterns in the sample representative of patterns in a population?
• Can we reject the possibility of independence? (The likelihood that there is no relationship)
• Is there a high or low probability of unrepresentativeness of findings?
• For example: Could patterns be caused by sampling errors?
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Document Summary
Quantitative data analysis (multivariate analysis and introduction to inferential statistics) If the rows total 100 percent each, it has been percentaged across: the rule, then, is as follows: If the table is percentaged down, read across. If the table is percentaged across, read down: when analyzing the table, remember to compare row percentages up and down within a column. In descriptive statistics a chi-square statistical test is commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. If the statistical test shows that a result falls outside the 95% region, you can be 95% certain that the result was not due to random chance, and is a significant result. Level of education of soldiers: acceptance at being inducted, their frie(cid:374)d(cid:859)s e(cid:454)pe(cid:396)ie(cid:374)(cid:272)es. This supports the idea that the original, zero- order relationship is genuine.
