Test of Hypothesis

Being confident make one's self more reassured. Briefly, explanations below are for two sided confidence levels/intervals in order to simplify the idea. Saying " two sided " gives initial impression that there is something like two limits, yeah they are: upper and lower limits where the confidence interval lies in between. Example: Let's look at the population of a specific mobile phone model. Suppose we are now interested in the ' weight ' property. We found that weight property follows a normal distribution with mean value of 120 grams and a standard deviation of 1.4 grams. Weight ~ Normal (Mu, Sigma) = Normal (120, 1.4) This understanding means that majority of mobiles tested will weigh very closely to 120 grams. Yes, there should be fluctuations above and below the mean value but surely that still relatively close to mean value. Suppose a question: do you expect weights like: 121, 119.5, 122.1, 118.9? Answer: Yes , I surely expect such

A general hypothesis is defined as following (eg a hypothesis on the population mean): H0: Mu = Mu0 H1: Mu !=  Mu0 OK, apart from we have a two or one sided hypothesis, after performing the checking and statistical tests: our conclusion should be one of the following: Rejecting the null hypothesis (H0). Failing to reject the null hypothesis (H0). The following statements for conclusions are not accurate : Accepting the null hypothesis (H0). Accepting the alternative hypothesis (H1). But why? When we fail to reject H0, it does not mean we accept H0 as a fact because we still could not prove it as a fact. But what happened is that we failed to prove it to be false. This goes like following: we have suspected new factors may affected the population mean, then we have taken all possible evidences and checking, but all checking failed to prove our suspects. As well, rejecting H0 does not mean accepting H1 as a fact. What happens in this case is we p

Fine... Constructing a statistical hypothesis is mainly to define what's called the null and the alternative hypotheses. Mostly in academic life, students are given the hypothesis to test. But in research or real experiment, constructing the hypotheses correctly is such vital step toward inference or statistical decision. Let's focus on hypotheses for population mean, for simplicity goals... Null hypothesis is mainly our initial information or primary belief about the population. Let's consider the production of 500 ml bottles. The 500 ml is the mean capacity of bottles capacity. Since this is the information we know from previous knowledge, it will be our null hypothesis. We write: H0: Mu = 500 Note: null hypothesis always has equality sign! Test for a hypothesis is usually done when we are worried that some factors affected the population, or population has changed for any possible reasons. OK, so the null hypothesis is the information we know primarily

Cool, say now we have a huge population with characteristics ( Mu, Sigma^2 ). When doing a study by sampling, we take a random sample ( size n items ) and then perform the study on the sample and conclude results back for the population. From Central Limit Theorem, we know that the sample mean will always follow a normal distribution  apart from what the population distribution is, such that: x_bar ~ N (Mu, Sigma^2/n) or say: Expected (x_bar) = Mu Variance (x_bar) = Sigma^2/n Well, let's see a simple illustrating example: Suppose we have a population with mean Mu=100 . Now, we have taken a sample, and computed the sample mean, x_bar. We mostly will have x_bar near 100 but not exactly 100. OK, let take another 9 separate samples... suppose these results: First sample --> x_bar = 99.8 Second sample --> x_bar = 100.1 .. .. .. 10th sample --> x_bar = 100.3 What we see that the sample mean is usually close to real population mean, that

A good fact to submit is that we can't easily know the exact truth values/parameters of a population. Mostly, population parameters also change slightly by time and/or affected by different surrounding factors. Example: a production line for the 500 ml bottles is assumed to produce a population of bottles such that mean value of bottles capacity is exactly 500 ml. Nice, but what happens in realty? In realty, several factors will mostly affect the production: human factors, machine factors, environment temperature...etc. Also, each new bottle will contribute in the population mean value. This means a continuous slight change, either up or down, of the mean capacity. Here comes the hypothesis! As you see, the ground truth value for population mean is difficult to be exactly determined. However, we have general assumptions/expectations. OK, constructing a hypothesis should always be driven by our initial knowledge and expectations about the population. Tes

You mostly know: the Standard Normal Distribution is the special case of Normal Distribution, given that: Mean: Mu=0.0 Variance: Sigma^2=1.0 Cool, as shown below: the family of normal distributions mainly vary in their mean value and/or their variance. The standard one plays the role of being the reference distribution. Well, we can convert any normal random variable to corresponding interpretations in the standard form. Hence, we simplify different computations only using standard normal distribution. OK, let's assume X is a normal distribution with mean Mu and variance Sigma^2. We can convert to a standard normally distributed random variable by following: Z=(X-Mu)/Sigma Here, we got Z as the standard normal distribution. OK, but what does this mean? Any point in X (with Mu, Sigma ) can be dealt exactly as the converted point in Z with mean Mu=0.0 and Sigma=1.0 . The numerator means: how much is the distance or difference between X and th