Learn Statistics

🧠 1. Introduction to Probability   🔹 What is Probability?  Probability is a way of measuring how likely an event is to happen. It's used when there is uncertainty about the outcome.   🔹 Probability Space A probability space includes:   Sample space (Ω): All possible outcomes of an experiment. E.g., Ω = {infected, not infected} for testing a cow for a disease.   Event space (𝓕): Collection of events we are interested in. E.g., event E = {infected}.    Probability (P): A number between 0 and 1 assigned to each event.  E.g., P(E) = 0.3 means a 30% chance of infection.   🧪 Example: Let’s say in a herd of 100 animals, 25 have a parasitic infection. P(infection) = 25/100 = 0.25   🔄 2. Conditional Probability  🔹 What is it?  Conditional probability is the chance of one event happening given that another has already occurred.    📌 Formula: 𝑃 ( 𝐴 ∣ 𝐵 ) = 𝑃 ( 𝐴 ∩ 𝐵 ) 𝑃 ( 𝐵 ) P(A∣B...

  I. Introduction to Confidence Intervals: A confidence interval is a range of values that is likely to include the true parameter of interest. In the context of estimating the mean, it provides a range within which we are reasonably confident the true population mean lies. Confidence intervals are useful because they convey both point estimates and a measure of uncertainty. II. Basics of Confidence Intervals for the Mean: Formula for Confidence Interval: � ˉ ± � ( � � ) x ˉ ± t ( n ​ s ​ ) � ˉ x ˉ is the sample mean. � t is the critical t-value from the t-distribution. � s is the sample standard deviation. � n is the sample size. III. The t-Distribution: The t-distribution is used when the sample size is small or when the population standard deviation is unknown. It is similar to the normal distribution but has fatter tails, accommodating the additional uncertainty associated with small sample sizes. IV. Calculating the Critical t-Value: The critical t-value is determined by...