Statistics and probability
Categories: Mathematics, Statistics
Course Content
Analyzing categorical data
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Analyzing one categorical variable
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Identifying individuals, variables and categorical variables in a data set
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Reading pictographs
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Reading bar graphs
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Reading bar graphs: Harry Potter
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Creating a bar graph
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Reading bar charts: comparing two sets of data
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Reading bar charts: putting it together with central tendency
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Reading pie graphs (circle graphs)
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Two-way relative frequency tables
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Interpreting two-way tables
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Categorical data example
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Analyzing trends in categorical data
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Two-way tables
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Two-way frequency tables and Venn diagrams
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Distributions in two-way tables
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Marginal and conditional distributions
Displaying and comparing quantitative data
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Displaying quantitative data with graphs
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Representing data
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Frequency tables & dot plots
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Creating a histogram
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Interpreting a histogram
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Stem-and-leaf plots
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Reading stem and leaf plots
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Describing and comparing distributions
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Shapes of distributions
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Clusters, gaps, peaks & outliers
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Comparing distributions with dot plots (example problem)
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Comparing dot plots, histograms, and box plots
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Example: Comparing distributions
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More on data displays
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Reading line graphs
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Misleading line graphs
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Measuring center in quantitative data
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Statistics intro: Mean, median, & mode
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Mean, median, & mode example
Summarizing quantitative data
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More on mean and median
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Comparing means of distributions
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Means and medians of different distributions
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Impact on median & mean: removing an outlier
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Impact on median & mean: increasing an outlier
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Missing value given the mean
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Median & range puzzlers
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Interquartile range (IQR)
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Interquartile range (IQR)
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Variance and standard deviation of a population
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Measures of spread: range, variance & standard deviation
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Variance of a population
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Population standard deviation
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Mean and standard deviation versus median and IQR
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Statistics: Alternate variance formulas
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Variance and standard deviation of a sample
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Sample variance
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Sample standard deviation and bias
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More on standard deviation
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Why we divide by n – 1 in variance
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Simulation showing bias in sample variance
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Simulation providing evidence that (n-1) gives us unbiased estimate
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Review and intuition why we divide by n-1 for the unbiased sample variance
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Box and whisker plots
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Worked example: Creating a box plot (odd number of data points)
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Worked example: Creating a box plot (even number of data points)
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Constructing a box plot
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Reading box plots
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Interpreting box plots
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Judging outliers in a dataset
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Other measures of spread
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Range and mid-range
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Mean absolute deviation (MAD)
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Mean absolute deviation example
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Percentiles
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Calculating percentile
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Analyzing a cumulative relative frequency graph
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Z-scores
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Normal distribution problem: z-scores (from ck12.org
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Comparing with z-scores
Modeling data distributions
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Effects of linear transformations
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How parameters change as data is shifted and scaled
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Density curves
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Density Curves
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Median, mean and skew from density curves
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Density curve worked example
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Worked example finding area under density curves
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Normal distributions and the empirical rule
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Qualitative sense of normal distributions (from ck12.org)
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Normal distribution problems: Empirical rule (from ck12.org)
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Standard normal distribution and the empirical rule (from ck12.org)
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More empirical rule and z-score practice (from ck12.org)
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Normal distribution calculations
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Standard normal table for proportion below
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Standard normal table for proportion above
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Standard normal table for proportion between values
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Finding z-score for a percentile
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Threshold for low percentile
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Normal distribution excel exercise
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More on normal distributions
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Deep definition of the normal distribution
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Introduction to scatterplots
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Constructing a scatter plot
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Example of direction in scatterplots
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Scatter plot: smokers
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Bivariate relationship linearity, strength and direction
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Correlation coefficients
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Example: Correlation coefficient intuition
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Calculating correlation coefficient r
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Introduction to trend lines
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Fitting a line to data
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Estimating the line of best fit exercise
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Estimating with linear regression (linear models)
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Line of best fit: smoking in 1945
Exploring bivariate numerical data
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Least-squares regression equations
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Introduction to residuals and least squares regression
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Calculating residual example
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Calculating the equation of a regression line
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Interpreting slope of regression line
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Interpreting y-intercept in regression model
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Interpreting a trend line
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Interpreting computer regression data
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Impact of removing outliers on regression lines
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Assessing the fit in least-squares regression
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Residual plots
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R-squared or coefficient of determination
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Standard deviation of residuals or Root-mean-square error (RMSD)
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More on regression
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Squared error of regression line
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Proof (part 1) minimizing squared error to regression line
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Proof (part 2) minimizing squared error to regression line
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Proof (part 3) minimizing squared error to regression line
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Proof (part 4) minimizing squared error to regression line
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Regression line example
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Second regression example
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Calculating R-squared
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Covariance and the regression line
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Statistical questions
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Statistical questions
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Statistical and non statistical questions
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Sampling and observational studies
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Reasonable samples
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Identifying a sample and population
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Examples of bias in surveys
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Example of undercoverage introducing bias
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Correlation and causality
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Sampling methods
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Picking fairly
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Techniques for generating a simple random sample
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Techniques for random sampling and avoiding bias
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Types of studies (experimental vs. observational)
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Types of statistical studies
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Worked example identifying experiment
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Worked example identifying observational study
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Worked example identifying sample study
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Appropriate statistical study example
Study design
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Experiments
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Introduction to experiment design
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Matched pairs experiment design
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Basic theoretical probability
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Intro to theoretical probability
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Simple probability: yellow marble
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Simple probability: non-blue marble
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Intuitive sense of probabilities
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The Monty Hall problem
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Probability using sample spaces
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Probability with counting outcomes
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Example: All the ways you can flip a coin
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Die rolling probability
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Subsets of sample spaces
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Basic set operations
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Intersection and union of sets
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Relative complement or difference between sets
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Universal set and absolute complement
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Subset, strict subset, and superset
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Bringing the set operations together
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Experimental probability
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Experimental probability
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Theoretical and experimental probabilities
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Making predictions with probability
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Randomness, probability, and simulation
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Experimental versus theoretical probability simulation
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Random number list to run experiment
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Random numbers for experimental probability
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Statistical significance of experiment
Probability
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Addition rule
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Probability with Venn diagrams
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Addition rule for probability
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Free-throw probability
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Three-pointer vs free-throw probability
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Probability without equally likely events
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Independent events example: test taking
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Die rolling probability with independent events
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Multiplication rule for independent events
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Sample spaces for compound events
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Compound probability of independent events
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Probability of a compound event
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Coin flipping probability
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Multiplication rule for dependent events
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Dependent probability introduction
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Dependent probability: coins
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Dependent probability example
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Independent & dependent probability
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Dependent probability
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Conditional probability and independence
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Calculating conditional probability
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Conditional probability explained visually
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Conditional probability tree diagram example
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Conditional probability and independence
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Analyzing event probability for independence
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Counting principle and factorial
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Count outcomes using tree diagram
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Counting outcomes: flower pots
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Permutations
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Permutation formula
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Zero factorial or 0!
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Factorial and counting seat arrangements
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Possible three letter words
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Ways to arrange colors
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Ways to pick officers
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Combinations
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Intro to combinations
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Combination formula
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Handshaking combinations
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Combination example: 9 card hands
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Combinatorics and probability
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Probability using combinations
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Probability & combinations (2 of 2)
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Example: Different ways to pick officers
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Example: Combinatorics and probability
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Getting exactly two heads (combinatorics)
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Exactly three heads in five flips
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Generalizing with binomial coefficients (bit advanced)
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Example: Lottery probability
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Conditional probability and combinations
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Mega millions jackpot probability
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Birthday probability problem
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Discrete random variables
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Random variables
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Discrete and continuous random variables
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Discrete and continuous random variables
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Constructing a probability distribution for random variable
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Probability models example: frozen yogurt
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Valid discrete probability distribution examples
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Probability with discrete random variable example
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Mean (expected value) of a discrete random variable
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Variance and standard deviation of a discrete random variable
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Continuous random variables
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Probability density functions
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Probabilities from density curves
Counting, permutations, and combinations
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Transforming random variables
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Impact of transforming (scaling and shifting) random variables
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Example: Transforming a discrete random variable
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Combining random variables
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Mean of sum and difference of random variables
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Variance of sum and difference of random variables
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Intuition for why independence matters for variance of sum
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Deriving the variance of the difference of random variables
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Example: Analyzing distribution of sum of two normally distributed random variables
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Example: Analyzing the difference in distributions
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Graphing basketball binomial distribution
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Binompdf and binomcdf functions
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Binomial random variables
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Binomial variables
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Recognizing binomial variables
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10% Rule of assuming “independence” between trials
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Binomial distribution
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Visualizing a binomial distribution
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Binomial probability example
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Generalizing k scores in n attempts
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Free throw binomial probability distribution
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Binomial mean and standard deviation formulas
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Mean and variance of Bernoulli distribution example
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Bernoulli distribution mean and variance formulas
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Expected value of a binomial variable
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Variance of a binomial variable
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Finding the mean and standard deviation of a binomial random variable
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Geometric random variables
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Geometric random variables introduction
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Probability for a geometric random variable
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Cumulative geometric probability (greater than a value)
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Cumulative geometric probability (less than a value)
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TI-84 geometpdf and geometcdf functions
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Proof of expected value of geometric random variable
Random variables
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More on expected value
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Term life insurance and death probability
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Getting data from expected value
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Expected profit from lottery ticket
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Expected value while fishing
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Comparing insurance with expected value
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Law of large numbers
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Poisson distribution
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Poisson process 1
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Poisson process 2
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What is a sampling distribution?
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Introduction to sampling distributions
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Sample statistic bias worked example
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Sampling distribution of a sample proportion
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Sampling distribution of sample proportion part 1
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Sampling distribution of sample proportion part 2
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Normal conditions for sampling distributions of sample proportions
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Probability of sample proportions example
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Sampling distribution of a sample mean
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Inferring population mean from sample mean
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Central limit theorem
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Sampling distribution of the sample mean
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Sampling distribution of the sample mean 2
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Standard error of the mean
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Example: Probability of sample mean exceeding a value
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Introduction to confidence intervals
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Confidence intervals and margin of error
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Confidence interval simulation
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Interpreting confidence level example
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Estimating a population proportion
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Confidence interval example
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Margin of error 1
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Margin of error 2
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Conditions for valid confidence intervals
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Conditions for confidence intervals worked examples
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Critical value (z*) for a given confidence level
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Example constructing and interpreting a confidence interval for p
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Determining sample size based on confidence and margin of error
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Estimating a population mean
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Introduction to t statistics
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Simulation showing value of t statistic
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Conditions for valid t intervals
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Example finding critical t value
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Example constructing a t interval for a mean
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Confidence interval for a mean with paired data
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Sample size for a given margin of error for a mean
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More confidence interval videos
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T-statistic confidence interval
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Small sample size confidence intervals
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The idea of significance tests
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Simple hypothesis testing
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Idea behind hypothesis testing
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Examples of null and alternative hypotheses
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P-values and significance tests
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Comparing P-values to different significance levels
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Estimating a P-value from a simulation
Sampling distributions
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Error probabilities and power
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Introduction to Type I and Type II errors
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Type 1 errors
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Examples identifying Type I and Type II errors
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Introduction to power in significance tests
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Examples thinking about power in significance tests
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Calculating a z statistic in a test about a proportion
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Calculating a P-value given a z statistic
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Making conclusions in a test about a proportion
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Tests about a population proportion
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Constructing hypotheses for a significance test about a proportion
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Conditions for a z test about a proportion
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Tests about a population mean
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Writing hypotheses for a significance test about a mean
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Conditions for a t test about a mean
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When to use z or t statistics in significance tests
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Example calculating t statistic for a test about a mean
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Using TI calculator for P-value from t statistic
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Using a table to estimate P-value from t statistic
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Comparing P-value from t statistic to significance level
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Free response example: Significance test for a mean
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More significance testing videos
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Hypothesis testing and p-values
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One-tailed and two-tailed tests
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Z-statistics vs. T-statistics
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Small sample hypothesis test
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Large sample proportion hypothesis testing
Confidence intervals
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Comparing two proportions
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Comparing population proportions 1
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Comparing population proportions 2
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Hypothesis test comparing population proportions
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Comparing two means
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Statistical significance of experiment
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Statistical significance on bus speeds
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Difference of sample means distribution
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Confidence interval of difference of means
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Clarification of confidence interval of difference of means
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Hypothesis test for difference of means
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Inference about slope
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Introduction to inference about slope in linear regression
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Conditions for inference on slope
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Confidence interval for the slope of a regression line
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Calculating t statistic for slope of regression line
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Using a P-value to make conclusions in a test about slope
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Using a confidence interval to test slope
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Nonlinear regression
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Comparing models to fit data example
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Transforming nonlinear data
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Worked example of linear regression using transformed data
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Analysis of variance (ANOVA)
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ANOVA 1: Calculating SST (total sum of squares)
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ANOVA 2: Calculating SSW and SSB (total sum of squares within and between)
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ANOVA 3: Hypothesis test with F-statistic
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