Statistics Concepts and Measures

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Questions and Answers

What does statistical significance primarily indicate?

  • The magnitude of a relationship in the data.
  • The practical importance of a finding.
  • The direction of a causal relationship.
  • The likelihood that the observed results are due to chance. (correct)

Which of the following describes a parameter?

  • A random selection of units of analysis.
  • A descriptive measure of a population. (correct)
  • A measure of statistical significance.
  • A characteristic calculated from a sample.

What does a positive correlation between two variables indicate?

  • As one variable increases, the other also tends to increase. (correct)
  • There is no relationship between the two variables.
  • As one variable increases, the other decreases.
  • The variables move in opposite directions.

Under what circumstance is a ‘simple random sample’ achieved?

<p>When each unit in the population has an equal chance of being selected. (A)</p> Signup and view all the answers

What is the Pearson's product moment correlation coefficient commonly denoted by?

<p>r (D)</p> Signup and view all the answers

In the regression equation $y = \alpha + \beta(x_i) + \epsilon$, what does $\beta$ represent?

<p>The average change in Y associated with a unit change in X. (C)</p> Signup and view all the answers

What does the law of large numbers state about sample statistics?

<p>As the number of observations increases, sample statistics get closer to the population parameter. (B)</p> Signup and view all the answers

What is a key requirement for making sound statistical inferences?

<p>A sufficiently sized sample selected randomly. (B)</p> Signup and view all the answers

What does a negative correlation between two variables suggest?

<p>As one variable increases, the other decreases. (C)</p> Signup and view all the answers

How is probability defined in a formal model of uncertainty?

<p>A numerical measure of the likelihood of an event occurring. (C)</p> Signup and view all the answers

What does 'mutual exclusivity' between events imply?

<p>If one event occurs, the other cannot occur. (D)</p> Signup and view all the answers

In the regression equation $y = \alpha + \beta(x_i) + \epsilon$, which component represents the error of trying to match the regression analysis to the real world?

<p>$\epsilon$ (D)</p> Signup and view all the answers

What effect does conditionality have on event probability?

<p>The probability of an event is influenced by whether a related event has occurred. (B)</p> Signup and view all the answers

What is a key difference between correlation and regression?

<p>Correlation measures the strength of any relationship, while regression shows the variable-specific relationship. (C)</p> Signup and view all the answers

What is the 'slope' of the regression line also known as?

<p>Regression coefficient (B)</p> Signup and view all the answers

What does it mean when we say two events are independent?

<p>The occurrence of one event has no effect on the probability of the other event. (A)</p> Signup and view all the answers

What is the primary purpose of classical hypothesis testing?

<p>To assess whether there is sufficient statistical evidence to reject a null hypothesis (A)</p> Signup and view all the answers

In hypothesis testing, what is the null hypothesis?

<p>A statement of no relationship or effect among variables. (D)</p> Signup and view all the answers

What does it mean to 'reject the null hypothesis'?

<p>There is sufficient evidence to support a claim by the data in the population. (D)</p> Signup and view all the answers

What is substantive significance primarily concerned with?

<p>The magnitude and direction of a statistical relationship. (C)</p> Signup and view all the answers

Under what condition is the $\chi^2$ test an appropriate method of statistical testing?

<p>When making inferences about relationships between nominal or ordinal variables. (D)</p> Signup and view all the answers

In the context of a $\chi^2$ test, what does a larger $\chi^2$ value indicate?

<p>The observed values greatly differ from the expected values. (C)</p> Signup and view all the answers

What is the purpose of degrees of freedom in statistical testing?

<p>To select the appropriate critical value for hypothesis testing. (D)</p> Signup and view all the answers

What does the standard error measure in the context of sampling distributions?

<p>The dispersion of the sample means around the population mean. (A)</p> Signup and view all the answers

What is the role of the t-value in a t-distribution?

<p>To define the critical region based on degrees of freedom and confidence level. (C)</p> Signup and view all the answers

What does a t-statistic (or t-score) represent?

<p>A test statistic from the sample for hypothesis testing. (A)</p> Signup and view all the answers

In a difference of means test, what is the null hypothesis typically?

<p>The difference between the means is zero. (B)</p> Signup and view all the answers

In a multiple regression, what do the regression coefficients estimate?

<p>The change in the dependent variable with a one unit change in the independent variable. (A)</p> Signup and view all the answers

What does the term 'alpha' ($\alpha$) represent in statistical hypothesis testing?

<p>The level of statistical significance, indicating the probability of rejecting the null hypothesis when it is in fact true. (A)</p> Signup and view all the answers

What is the interpretation of the intercept in a multiple regression equation?

<p>The predicted value of Y when all independent variables are zero. (B)</p> Signup and view all the answers

What is the practical purpose of a test statistic in hypothesis testing?

<p>To indicate the degree to which the sample result deviates from what you would expect if the null hypothesis was true. (A)</p> Signup and view all the answers

What is the role of the critical value in statistical hypothesis testing?

<p>It defines the boundaries of the rejection region which are determined by both the confidence level and the degrees of freedom. (B)</p> Signup and view all the answers

Why are multiple regression coefficients referred to as 'partials'?

<p>They represent each variable's unique contribution, controlling for other variables. (B)</p> Signup and view all the answers

What does the p-value in multiple regression represent?

<p>The probability that the sample regression coefficient is random. (B)</p> Signup and view all the answers

Which of the following is NOT a property of a normal distribution?

<p>It is asymmetric. (D)</p> Signup and view all the answers

How do Z-scores standardize variables?

<p>By expressing values in terms of how many standard deviations they are from the mean. (A)</p> Signup and view all the answers

According to the Central Limit Theorem, what happens to the distribution of sample means as the sample size increases?

<p>It approaches a normal distribution, regardless of the population distribution. (D)</p> Signup and view all the answers

What is the purpose of constructing a confidence interval?

<p>To provide an estimate of a population parameter within a range, with an associated confidence level. (B)</p> Signup and view all the answers

What does a 95% confidence interval suggest about the population parameter?

<p>There is a 95% probability that the population parameter is within the interval. (D)</p> Signup and view all the answers

What do predicted probabilities represent in a statistical model?

<p>The probabilities of the dependent variable at different values of the independent variable(s). (B)</p> Signup and view all the answers

In the context of statistical modeling, what are ideal types?

<p>Profiles that summarize the marginal effects of key variables. (A)</p> Signup and view all the answers

What is the purpose of the Likelihood Ratio chi2 test?

<p>To determine whether a model is specified correctly. (C)</p> Signup and view all the answers

What does the Wald test primarily assess in statistical models?

<p>Whether regression coefficients are equal to zero, another number, or each other. (B)</p> Signup and view all the answers

What do odds ratios generally indicate?

<p>The odds of y being equal to 1 over y equal to 0. (A)</p> Signup and view all the answers

In ordinal logistic regression, what does the proportional odds assumption state?

<p>The intervals between each adjacent outcome in the dependent variable are uniform. (D)</p> Signup and view all the answers

How does a multinomial logit model estimate the model with a nominal dependent variable?

<p>By estimating binary logits for each outcome category against a base category (B)</p> Signup and view all the answers

What are 'Cutpoints' in the context of ordinal logistic regression?

<p>The points when the probability of the impact of values of independent variables changes for outcome categories of the ordinal-level dependent variable. (C)</p> Signup and view all the answers

Flashcards

What is a parameter?

A descriptive characteristic of a population.

What is a statistic?

A statistical value calculated from sample data. It's used to estimate an unknown population parameter.

What is a simple random sample?

Each unit in the population has an equal chance of being selected for the sample.

What is statistical significance?

When the results observed in a sample are highly unlikely to be due to random chance, indicating a real effect or relationship.

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What is substantive significance?

The size or magnitude of a description, relationship, or pattern found in a study.

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What is randomization?

The selection of units of analysis based on chance, not design.

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What is probability?

A formal model of uncertainty that assigns numerical values to the likelihood of events occurring. It helps quantify the chances of different outcomes.

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What is conditional probability?

The probability of an event occurring when another related event has already happened.

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Pearson's Product Moment Correlation

The most commonly used correlation measure, calculated using interval-level variables.

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Correlation

A statistical measure that describes the strength and direction of the linear relationship between two interval-level variables. Positive correlation indicates both variables increase or decrease together, while negative correlation means they move in opposite directions.

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Regression Analysis

A statistical method used to examine the linear relationship between a dependent variable and an independent variable. It provides a mathematical equation that describes how changes in the independent variable influence the dependent variable.

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Regression Equation

A mathematical equation representing the linear relationship between a dependent variable (Y) and an independent variable (X). It consists of an intercept, slope, and error term.

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Bivariate Linear Regression

A type of regression analysis that examines the relationship between two interval-level variables. It provides information on how these variables move together, respecting the theoretical order of the independent and dependent variables.

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Regression Coefficient

A component of the regression equation that represents the average change in the dependent variable associated with a unit change in the independent variable. It describes both the magnitude and direction of the relationship.

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Intercept

A component of the regression equation that represents the value of the dependent variable when the independent variable is zero. It is where the regression line intersects the Y-axis.

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Error Term

A component of the regression equation that accounts for the random variations or errors in the relationship between the dependent and independent variables.

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Independent events

Two events are independent if the outcome of one event does not affect the probability of the other event occurring.

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Classical hypothesis testing

A statistical procedure used to determine if there is enough evidence to reject a null hypothesis.

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Null hypothesis

A statement that there is no relationship or difference between variables. It is the opposite of the alternative hypothesis.

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Alternative hypothesis

A statement that predicts a relationship or difference between variables. It is the main idea you are trying to prove.

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Substantive significance

The extent to which the results of a study are meaningful and relevant in the real world.

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Statistical significance

The confidence in which the results of a study can be generalized to a larger population.

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Chi-square test (χ²)

A statistical test used to determine if there is a significant relationship between two or more categorical variables.

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Degrees of freedom

The number of independent pieces of information used to calculate a statistic.

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Alpha (α)

The probability of making a Type I error (rejecting the null hypothesis when it's true). It represents the maximum risk of incorrectly concluding there's an effect when there isn't one. Common values are 0.05 (95% confidence), 0.01 (99% confidence), and 0.001 (99.9% confidence).

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Test statistic

A numerical value calculated from sample data that quantifies the deviation from the expected value. It's used to judge how likely our observed result is due to chance.

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Confidence interval

A range of values around a sample estimate that we expect to contain the true population parameter with a certain level of confidence. It accounts for the uncertainty in our sample and helps us make inferences about the population.

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Critical value

A value that defines the boundary of the critical region in hypothesis testing. It's determined by the chosen confidence level and the sample's degrees of freedom. If the test statistic falls within this region, we reject the null hypothesis.

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Statistical inference

The process of using sample data to make inferences about population parameters. This involves estimating population characteristics or testing hypotheses about them.

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Normal distribution

A symmetrical probability distribution shaped like a bell curve. It's used to model many natural phenomena and is crucial in statistical inference due to its predictable properties.

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Central Limit Theorem

A powerful concept in statistics that states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population distribution.

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Standardizing variables

The process of transforming variables with different units and scales to a common standardized score. This allows for direct comparison of different variables.

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Standard Error

A measure of dispersion used for sampling distributions, indicating how much sample means vary around the population mean.

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t-Distribution

A statistical distribution similar to the normal distribution but accounts for sample size, particularly when dealing with small samples.

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t-Value

A value from the t-distribution that defines the critical region for hypothesis testing, depending on the degrees of freedom and the chosen confidence level.

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t-Test

A statistical test used to compare the means of two groups, based on the t-distribution.

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Multiple Regression

A statistical technique to analyze the relationship between a dependent variable and multiple independent variables simultaneously.

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Adjusted R^2

The measure of explained variance in the dependent variable by the set of independent variables, adjusted for the number of variables and observations.

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Partial Regression Coefficients

Regression coefficients in multiple regression represent the partial contribution of each independent variable to the total explained variance, controlling for the other variables.

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p-value in Multiple Regression

The probability that the estimated regression coefficient is due to chance, calculated for each independent variable in multiple regression.

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Predicted Probabilities

The probability of the dependent variable taking on a specific value, given different values of the independent variables.

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Likelihood Ratio Chi2 Test

A statistical method used to test if the regression model is correctly specified. It checks for any deviations in the assumed relationships.

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Ideal Types

A summary of the effects of key variables, grouped into distinct profiles that are relevant to the research question.

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Wald Test

A statistical test used to compare different models where regression coefficients are assumed to be either equal to zero, a specific value, or to each other.

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Odds Ratio

The odds of the dependent variable being equal to 1 (often a 'success') compared to the odds of it being equal to 0 (often a 'failure').

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Ordinal Regression Model

A type of regression model used when the dependent variable has more than two outcomes, but these outcomes have a natural order or ranking.

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Proportional Odds Assumption

The assumption that the differences between adjacent categories in the dependent variable are constant across all values of the independent variables.

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Multinomial Logit Model

A regression model designed for categorical dependent variables with more than two categories that are not ordered (nominal level).

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Study Notes

Univariate Descriptive Statistics

  • A population is the entire group of interest
  • A sample is a subset of the population
  • Conceptualization is defining the concepts of interest
  • Operationalization is measuring the concepts empirically
  • Level of measurement describes the type of data:
    • Nominal: unordered categories (e.g., colors)
    • Ordinal: ordered categories (e.g., ratings)
    • Interval: ordered with equal intervals (e.g., temperature)
  • Measures of central tendency:
    • Mode: most frequent category (nominal data)
    • Median: middle value (ordinal data)
    • Mean: average (interval data)
  • Measures of dispersion:
    • Variance: how spread out the data are around the mean
    • Standard deviation: average distance from the mean
    • Range: difference between highest and lowest values
    • Relative frequencies: proportion of each category
    • Variation ratio: measure of variability around the mode (nominal data)

Measures of Association

  • Cross-tabulation (cross-tab): displays joint distribution of nominal/ordinal variables
  • Joint distribution: distribution of responses as a function of another variable
  • Yule's Q: 2x2 form of Goodman and Kruskal's gamma for nominal/ordinal data
  • Lambda: proportional reduction of error (PRE) measure for nominal by ordinal variables
  • Goodman and Kruskal's Gamma: PRE measure for ordinal variables
    • Positive relationship: variables increase/decrease together
    • Negative relationship: variables move in opposite directions
  • Zero-order relationship: relationship between two variables only

Bivariate Measures of Association

  • Means comparison: compares means of interval variables across categories of a nominal/ordinal variable
  • Scatterplot: graph of joint distribution of two interval variables
  • Correlation: measures the extent of linear relationship between two interval variables
    • Positive correlation: variables increase/decrease together
    • Negative correlation: variables move in opposite directions (one increases, other decreases)
  • Regression analysis: describes linear relationship between dependent and independent variable

Bivariate Regression

  • Bivariate linear regression: relationship between two interval variables
  • Regression equation: Y = α + βX₁ + ε
    • Y: dependent variable
    • X₁: independent variable
    • α: intercept
    • β: regression coefficient
    • ε: error term
  • Regression coefficient (β): average change in Y for a unit change in X
  • Intercept (α): value of Y when X = 0
  • Error term (ε): unexplained variation in Y

Inference for Nominal and Ordinal Data

  • Statistical significance: unlikely that results are due to chance
  • Substantive significance: practical importance of the results
  • Parameter: descriptive characteristic of a population
  • Statistic: estimate of a parameter from sample data
  • Alpha: significance level (probability of incorrectly rejecting the null hypothesis) → 0.05 (95% confidence) or 0.01 (99%) is commonly used.
  • Confidence intervals: range of plausible values for a population parameter
  • Probability: numerical measure of event likelihood
  • Hypothesis testing: step-by-step procedure to determine statistical significance
  • Null hypothesis (H₀): no relationship/difference
  • Alternative hypothesis (H₁): there is a relationship/difference

Inference for Interval Data

  • Model fit: appropriateness of included independent variables
  • Coefficient of determination (R²): proportion of common variation between variables
  • Central Limit Theorem: sample means approach a normal distribution as sample size increases.

Multiple Regression

  • Multiple regression: tests several independent variables simultaneously; controls for the impact of each variable on the dependent variable.
  • Independent variable (X): variable influencing Y
  • Dependent variable (Y): variable being influenced
  • Regression coefficients (β): average change in Y related to a unit change in X

Dummy Variables

  • Dichotomous variables to include nominal/ordinal data
  • Reference category: against which all other dummy variables are compared

Binary Logistic Regression

  • Determines the probability of a dependent variable taking on a value of 1
  • Categorical dependent variables

Ordinal Logistic Regression

  • Handles ordinal dependent variables
  • Proportional odds assumption: intervals between adjacent outcomes are uniform

Truncated and Censored Data

  • Truncated: observations systematically excluded
  • Censored: observations with unknown values beyond a terminal value
  • Count data: occurrences of an event within a fixed period

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