Inferential Statistics

Satistical Inferences

Population: The population is a collection of objects, both animate and inanimate, that is being studied.
Statistical individuals refer to objects in a population. The population can be finite or infinite depending on the number of things in the population.

Sample: A sample is a finite subset of statistical individuals (objects) in a population.
Sample Size: The number of statistical individuals (objects) in a sample is called the sample size.

Parameters: The parameters of the population are statistical constants or measures of the population, such as mean(μ), variance(σ²), etc. 

Statics: Statistics are statistical measurements or constants computed only from sample observations, such as mean$\left(\overline{X}\right)$, variance(s²) and so on.

SAMPLING DISTRIBUTION

If we take 50 random samples from a population and calculate their means, we'll end up with a sequence of 50 means that make up a frequency distribution. The sampling distribution of means is the term given to this distribution.
In general, if S₁, S₂, S₃, ...., S_n are values of a static S(mean, variance, etc.) obtained from n independent random samples of a definite size chosen from a given population, then S₁, S₂, S₃, ...., S_n form a sampling distribution of statistics S. The mean ($\overline{S}$) and variance of statistic S are given by $\overline{S}=\frac{1}{n}\sum _{i=1}^n{S}_i \mathrm{and} Var\left(S\right)=\frac{1}{n}\sum _{i=1}^n{\left(S_I-\overline{S}\right)}^2$.
Standard Error(S.E.): The standard error of a statistic is the standard deviation of its sampling distribution.

STATISTICAL INFERENCES

Statistical inference's main goal is to make inferences about a population parameter based on the examination of a sample selected from that population. 
Static inference consists of 2 major areas:
(i) estimation
(ii) testing of hypothesis

The null hypothesis, which we make regarding a population parameter, is the starting point for hypothesis testing. The null hypothesis states that no substantial difference exists between the sample statistics and the corresponding population parameter, or between the two sample statistics. The null hypothesis is usually denoted by H₀. 
An alternate hypothesis is any hypothesis that is in addition to the null hypothesis, usually denoted by H₁.
When we test the null hypothesis that the population has a specified mean μ_o, i.e. H₀: μ = μ_o, then the alternative hypothesis could be:
(i) H₁: μ ≠ μ_o           (Two tailed alternative)
(ii) H₁: μ > μ_o          (Right tailed alternative)
(iii) H₁: μ < μ_o         (Left tailed alternative)

SIGNIFICANCE LEVEL

In a hypothesis test the significance level, α, is the probability of making a wrong decision when the null hypothesis is true.
For example significance level 0.05 indicates a 5% risk of concluding that a difference exists (between a population parameter and sample statistic or between statistics of two samples) when there is no actual difference.

To graph a significance level of 0.05 in the two-tailed distribution, we need to shade the 5% of the distribution, which is furthest away from the null hypothesis. In the graph below the two shaded areas are equidistant from the null hypothesis value and each has a probability of 0.025, for a total of 0.05. These shaded areas are called the critical regions for a two-tailed test.



CONFIDENCE…