The Fact and the Hypothesis

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.

Testing the hypothesis is statistical checking methods to judge these beliefs. Test results should conclude/push more beliefs into either:

Failure to say the parameter (eg mean value) has changed. Example: the 500 ml bottles capacity should be considered stable/no change at 500 ml. This conclusion is known as (failing to reject the null hypothesis).
Rejecting this initial assumption (we call it later the null hypothesis). This means that some factors affected the production and the mean value has changed to different values (up or down). Then, further monitoring/improvements should be decided to solve the issue. This conclusion is known as (rejecting the null hypothesis).

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The "Sample"

Anytime you aim to perform a study on the entire population, you will surely find that this task will be: Much time and/or efforts consuming as populations are normally huge . Impossible if the population is infinite (such as products). Here comes the role of taking samples. Yes! we just take a sample from the whole population, perform the study on the chosen sample, apply the results back to our population. This is the core of inferential statistics because what we do is to infer parameters/properties of the population using information from a small sample. Well, this does not mean we will obtain 100% exact accurate estimations or inferences. But to be as close as possible, sample elements should be taken randomly ! At least, being random in sample selection will mostly include the diversity of information/facts within our population.

Understanding the distribution of sample mean (x_bar)

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

Conclusions of Hypothesis Testing

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

Test of Hypothesis

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