INTRODUCTION TO INFERENTIAL STATISTICS

Inferential Statistics is one of the two main branches of statistics. It uses a sample from the population to draw inferences about the population.

POPULATION AND SAMPLE

What is Population? What is Sample?

Let us first understand the difference between Population and Sample. Population refers to a set of each and every element with some common characteristics. But, at times, it is impractical or impossible to study every member of the population. To overcome this, a group of members are selected to study. This group of members is referred to as Sample. This sample is then used to make inferences about the general population.

PARAMETER AND STATISTICS

Fixed measures that describe the population are called Parameters. Many a times, it is very difficult to estimate parameters, especially when the population is large. Thus we need to estimate them. This is when Statistics comes to the rescue. It is a measure that describes the sample. It can be calculated very easily given a sample at hand.

To sum up, Parameters are fixed and unknown and describe the population. Statistics are variable and known (as we can calculate it from the available sample) and describes the sample.

Note: Samples only give estimates of the population. Hence, there is a difference between sample statistics and population parameter. This difference in called the Sampling Error.

WHAT IS INFERENTIAL STATISTICS?

Inferential Statistics refers to the technique of estimating the population parameters from sample statistics. It tries to make conclusion about the population that is beyond the available data.

Unlike DESCRIPTIVE STATISTICS, this is purely based on the Theories of Probability and has an associated accuracy which tells us how much the sample is representative of the population.

Estimation and Hypothesis Testing are two major techniques of Inferential Statistics.

ESTIMATION

Estimation is a technique that uses the available data (sample) to infer the population parameter. There are two forms of Estimation.

1. Point Estimation: Point estimation calculates a single value that is the best estimate of the population parameter. This value is called point estimate. For example, sample mean is a point estimate of the population mean. An equation, which on substitution, gives point estimate, is called point estimator.
2. Interval Estimation: An interval estimator gives an interval in which the population parameter is likely to fall with some confidence. This interval is called interval estimate or confidence interval. Lets suppose, (a,b) is the 95% confidence interval of the population mean, this means that we can be 95% certain that the population mean lies in the interval (a,b).

HYPOTHESIS TESTING

What is a Hypothesis??

A hypothesis is a claim. It is just an assumption which can be proven to be false, but cannot be tested to be true. There are two types of hypothesis.

1. Null Hypothesis: A Null Hypothesis is a claim of no significant change or difference, which is tested for possible rejection. Usually, the null hypothesis is denoted by Ho.
2. Alternative Hypothesis: An alternative hypothesis is a claim of some difference from the null hypothesis. It is usually denoted by H1.

Type 1 and Type 2 Error:

Steps in Hypothesis Testing:

1. Select the parameter of interest.
2. State the Null Hypothesis and the Alternative Hypothesis.
3. Specify the distribution of the sample and the corresponding test statistics.
4. Set a maximum value of the Type 1 Error (which is also called alpha, or the level of significance).
5. Under the assumption that the Null Hypothesis is true, calculate the test statistics that will be used for testing.
6. Depending on the level of significance, find the critical value (to be compared with test statistics) or the p-value (to be compared with the level of significance).
7. Compare the calculated test statistics and the critical value. Alternatively, compare the p-value and the level of significance. Basis on this comparison, we either reject the Null Hypothesis, or we fail to reject the Null Hypothesis.

Note: The Null Hypothesis is rejected if the calculated test statistics is greater than the critical value. Alternatively, if the p-value is less than the level of significance, reject the Null Hypothesis.

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