AYRUN(Always Run)

Unit 16: Analysis of Variance (ANOVA) 1. What is ANOVA? Analysis of Variance (ANOVA) is a statistical method used to compare the means of three or more groups to determine if at least one group mean is different from the others. It helps answer questions like: "Do different teaching methods lead to different average test scores?" 2. Types of ANOVA One-way ANOVA: Compares means across one factor (e.g., test scores by teaching method). Two-way ANOVA: Compares means across two factors (e.g., test scores by teaching method and gender). 3. Assumptions of ANOVA Independence of observations Normality (data in each group are approximately normal) Homogeneity of variances (each group has similar variance) 4. The ANOVA Table and F-statistic ANOVA uses the F-statistic to compare the variance between group means to the variance within groups. The ANOVA table summarizes the calculations: Source Sum of Square...

Unit 15: Advanced Regression (Inference and Transforming) 1. Inference about Slope In regression analysis, the slope of the regression line represents the average change in the response variable for each one-unit increase in the explanatory variable. Inference about slope involves determining whether the observed relationship between variables is statistically significant or could have occurred by random chance. Null Hypothesis (H 0 ): The true slope is zero (no relationship). Alternative Hypothesis (H a ): The true slope is not zero (there is a relationship). We use a t-test to test the significance of the slope. The test statistic is: t = (b - 0) / SE b where b is the sample slope and SE b is its standard error. Quiz: Inference about Slope 1. What does the slope of a regression line represent? --Select-- The average change in the response variable for each unit increase in the expl...

Unit 14: Inference for Categorical Data (Chi-Square Tests) Topics Covered: Chi-square goodness-of-fit tests Chi-square tests for relationships 1. Chi-square Goodness-of-Fit Tests The chi-square goodness-of-fit test is used to determine whether a set of observed categorical data matches an expected distribution. It answers questions like: "Does a die appear to be fair?" or "Are the colors in a bag of candies distributed as claimed by the manufacturer?" Null hypothesis (H 0 ): The observed frequencies match the expected frequencies. Alternative hypothesis (H a ): The observed frequencies do not match the expected frequencies. Test statistic: \( \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \), where \(O_i\) is observed and \(E_i\) is expected count for category \(i\). Degrees of freedom: Number of categories minus 1. Take Goodness-of-Fit Quiz 2. Chi-square Tests for Relationships (Independ...

Unit 13: Two-sample inference for the difference between groups Comparing Two Proportions What is it? Comparing two proportions involves determining whether the proportion of a certain outcome is different between two groups. For example, you might want to know if the proportion of people who prefer tea is different between two cities. Null hypothesis (H 0 ): The proportions are equal (p 1 = p 2 ). Alternative hypothesis (H a ): The proportions are not equal (p 1 ≠ p 2 ), or one is greater/less than the other. Test statistic: Uses the difference between sample proportions and the standard error. Conditions: Random samples, independence, and large enough sample sizes (np ≥ 10 and n(1-p) ≥ 10 for both groups). Visualize Example Quiz: Comparing Two Proportions Suppose 60 out of 100 people in Group A like chocolate, and 45 out of 100 in Group B like chocolate. Is there evidence that the proportions are diffe...

Unit 12: Significance Tests (Hypothesis Testing) Topics Covered: The idea of significance tests Error probabilities and power Tests about a population proportion Tests about a population mean The Idea of Significance Tests Significance tests, also known as hypothesis tests, are statistical methods used to make decisions or inferences about population parameters based on sample data. The process involves: Stating a null hypothesis (H 0 ) and an alternative hypothesis (H a ). Collecting and summarizing sample data. Calculating a test statistic and a p-value. Comparing the p-value to a significance level (α) to decide whether to reject H 0 . Quiz: Error Probabilities and Power When conducting significance tests, two types of errors can occur: Type I Error (α): Rejecting th...

Unit 11: Confidence Intervals Master the concepts of confidence intervals with explanations, interactive quizzes, and visualizations. Introduction to Confidence Intervals A confidence interval is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. It is commonly used in statistics to estimate population parameters (like mean or proportion) and to express the uncertainty of this estimate. Confidence Level: The probability that the interval contains the true parameter (e.g., 95%). Margin of Error: The range above and below the sample statistic. Visualization: The blue bar is the confidence interval; the red line is the true parameter. Estimating a Population Proportion To estimate a population proportion (like the percentage of people who prefer tea over coffee), we use a sample proportion ( p̂ ) and construct a confidence interval...

Unit 10: Sampling Distributions What is a sampling distribution? A sampling distribution is the probability distribution of a given statistic based on a random sample. It shows how the value of a statistic (like the mean or proportion) varies from sample to sample, even when all samples are drawn from the same population. Sampling distributions are fundamental in inferential statistics because they allow us to estimate the variability of a statistic and make probability-based conclusions about a population. Sampling distribution of a sample proportion The sampling distribution of a sample proportion describes the distribution of sample proportions (\( \hat{p} \)) from all possible samples of the same size from a population. If the population proportion is \( p \) and the sample size is \( n \), the mean of the sampling distribution is \( p \) and the standard deviation is \( \sqrt{\frac{p(1-p)}{n}} \). For large enough samples, the distribution is approx...

Unit 9: Random Variables Topics: Discrete random variables Continuous random variables Transforming random variables Combining random variables Binomial random variables Binomial mean and standard deviation formulas Geometric random variables More on expected value Poisson distribution Discrete Random Variables A discrete random variable is a variable that can take on a countable number of possible values. Examples include the number of heads in 10 coin tosses, or the number of students present in a class. The probability distribution of a discrete random variable lists each possible value and its probability. Quiz: Continuous Random Variables A continuous random variable can take on any value within a given range. Examples include the height of students or the time it takes to run a rac...

Unit 8: Counting, Permutations, and Combinations Topics: Counting principle and factorial Permutations Combinations Combinatorics and probability Counting Principle and Factorial The counting principle helps us find the total number of possible outcomes by multiplying the number of choices for each event. For example, if you have 3 shirts and 2 pants, you have 3 × 2 = 6 possible outfits. The factorial of a number n (written as n!) is the product of all positive integers up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Quiz: What is 6! ? Check Permutations A permutation is an arrangement of objects in a specific order. The number of ways to arrange n objects is n! If you want to arrange r objects out of n, use the formula: P(n, r) = n! / (n - r)! Quiz: How many ways can you arr...