Title Sampling, Point Estimation and Interval Estimation.pdf ✅

0.1. Introduction to Statistics 1 0.1 Introduction to Statistics As the world grows in complexity, it becomes increasingly difficult to make informed and intelligent decisions. Often these decisions must be made with less than perfect knowledge and in the presence of some uncertainty. These decisions are made with the help of the results derived from statistics. Statistics is concerned with the development of methods and their applications for col- lecting, analyzing and interpreting quantitative data in such a way that the reliability of a conclusion based on data may be evaluated objectively by means of probability statements. Probability theory is used to evaluate the reliability of conclusions and inferences based on data. Thus, probability theory is fundamental to mathematical statistics. 0.2 Probability v/s Statistics Before we start with new concepts, we need to make a distinction between probability and statistics. Suppose We’re given a model, let’s say Bernoulli(p = 0.6) and we want to find the probability of some data. For example, given this model, what is the probability of T HT T HH, or P(T HT T HH)? That’s something you know how to do now! What we’re going to focus on now is going the opposite way. Given a coin with an unknown probability of heads, we flip it a few times and get THTHH. How can we use this data to predict/estimate this value of p? In probability theory, we already have a model, we generate data and calculate its prob- ability while in statistics, we observe the data that we have collected and try to build a model that fits it.
0.3. Sampling 2 Probability Statistics Given model, predict data Given data, predict model Predicted Model Data Assumed Model P{ } Figure 1: Probability versus statistics. 0.3 Sampling 0.3.1 Data Data Data are the facts and figures collected, analyzed, and summarized for presentation and interpretation. Different forms of data: 1. Quantitative Data: Quantitative (or numerical) data consist of numbers representing counts or measurements. Example: The age (in years) of Delhi residents, population of Meerut city, etc. 2. Categorical Data: Categorical (or qualitative or attribute) data consists of names or labels that are not numbers representing counts or measurements. Example: The numbers like 24, 28, 17, 07, and 31 written on the shirts of the Indian
0.3. Sampling 3 cricket players. These numbers are substitutes for names. They don’t count or measure anything, so they are categorical data. Quantitative Data can be categorised as: 1. Discrete data result when the number of possible values is either a finite number or a “countable” number. That is, the number of possible values is 0 or 1 or 2, and so on. Example: The numbers of eggs that hens lay are discrete data because they represent counts. 2. Continuous (numerical) data result from infinitely many possible values that corre- spond to some continuous scale that covers a range of values without gaps, interrup- tions, or jumps. Example: The amounts of milk from cows are continuous data because they are mea- surements that can assume any value over a continuous span. What is Data? Information Quantities graphs measurement observations facts numbers Figure 2: What is Data? 0.3.2 Population In statistics, the term ”population” refers to the entire group of observations or the dataset that completely characterises a phenomenon of interest that we want to study. It includes all the potential data points (observations) related to the phenomenon we’re interested in.
0.3. Sampling 4 The population is just the set of all possible outcomes in our sample space. It may be finite or infinite. We can define a sample as a subset of data selected from a population. Population A statistical population is a data set (usually large, sometimes conceptual) that is our target of interest. Gathering information from a whole population requires a census. Census A census is the collection of data from every member of the population. 0.3.3 Sample In answering the questions in the above example for a given population, we never give the answer using the entire population. The reasons being: 1. It is not always possible that we have data for the entire population. Also, data collection is a very costly process. It is easier and cost-effective to process a smaller subset of the population rather than the entire group. 2. It is much more practical to use a sample instead of the entire population due to its large size. Samples provide a representation of the entire population. 3. Obtaining results based on samples is time-effective, manageable and feasible. Sample A sample is a subset of data selected from the target population. We choose a sample to gather information and answer questions about the population.