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Inferential Statistics

Satistical Inferences


A factory unit producing cars chooses some for inspection and quality check by the quality control department. These cars are chosen randomly for the quality check, form a sample and the totality of the cars manufactured is called the population.
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.
Sampling is a common practice in our day-to-day lives. We use sampling to help us with any statistical inquiry. For example, by analyzing certain samples of specific objects, we may be able to decide whether or not to accept or reject them. The acceptance or rejection of a sample or samples depending on their qualities, on the other hand, results in a sampling error.

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 meanX, variance(s2) and so on.


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, ...., Sn 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, ...., Sn form a sampling distribution of statistics S. The mean (S) and variance of statistic S are given by S=1ni=1nSi  and VarS=1ni=1nSI-S¯2.
Standard Error(S.E.): The standard error of a statistic is the standard deviation of its sampling distribution.


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 μo, i.e. H0: μ = μo, then the alternative hypothesis could be:
(i) H1: μ ≠ μo           (Two tailed alternative)
(ii) H1: μ > μo          (Right tailed alternative)
(iii) H1: μ < μo         (Left tailed alternative)


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.


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