Applied Statistics: Inferential Statistics


Inferential statistics is something that is very important and essential for Machine Learning. Descriptive statistics gives an in-depth idea about the data, but the decision-making algorithms for Machine Learning are all built upon inferential statistics. Concept of Hypothesis testing is something that actually is helpful to predict future outcomes.


Before we dive deep into the Hypothesis testing let us understand some concepts:

Concept of Sampling Distribution:

To analyze the sample and make inferences about the population is the main purpose of Sampling Distribution.

Sampling Distribution: Distribution of a particular sample statistic of all possible samples that can be drawn from a population.

Central Limit theorem:

·         If n samples are drawn from a population that has a mean and standard deviation then the sampling distribution follows a normal distribution.

·         Corresponding Z-score transformation would be as shown below:



·         If the population is normal, this holds true even for smaller sample sizes.

·         However, if the population is not normal, this holds true for sufficiently large sample sizes.

·         Sampling Distribution of the mean of any independent random variable will be normal.

·         This applies to both discrete and continuous distributions.

·         The random variable should have a well-defined mean and variance.

·          Applicable even when the original variable is not normally distributed

·         Assumption 1: The data must be randomly sampled

·         Assumption 2: The sample values must be independent of each other.

·         Assumption 3: The 10% condition: When the sample is drawn without replacement, the sample size n, should be no more than 10% of the population.

 

 

Hypothesis:

A Hypothesis is an unproven proposition or supposition that tentatively explains certain facts or phenomena.

For a population, a hypothesis can be very well defined as follows:


Let us consider an example of the formulation of the Hypothesis:

Coca-Cola’s most selling product is 600ml coke and since 600ml info is on the label we assume it to be true. But is it actually true?


Steps for this example in the formulation of the Hypothesis:

  1. Collect 100 bottles from all over the country, so that we have a random sample.
  2. Measure volume of each bottle in the sample to find the mean of 100 bottles.
  3. Use sample mean to test assumption (status quo, this being the key in making the inferences about the population from the sample).

When formulating Hypothesis:

Negative: Am I testing a status quo that already exists? This is the Null hypothesis, where there is the negation of the research question. Always contains equality, =, >=, <=

Positive: Am I testing an assumption or claim, that is something beyond what I know? This is the alternate hypothesis, where there is a research question to be proven. Doesn’t contain equality, !=, <, >

Null and Alternative Hypothesis: All statistical conclusions are made in reference to the null hypothesis. We either reject the null hypothesis or fail to reject the null hypothesis. We do not accept the null hypothesis. From the start, we assume the null hypothesis to be true, later the assumption is rejected or we fail to reject it.

When we reject the hypothesis, we can conclude that the alternative hypothesis is supported.

If we fail to reject the null hypothesis, it does not mean that we have proven the null hypothesis is true. Failure to reject the null hypothesis doesn’t equate to proving that it is true. It just holds up our assumptions or the status quo.

Type 1 and Type 2 errors:

H0 = Hypothesis





In simple terms, we can consider an example of a criminal. In the court of law, there is only a consideration if the criminal is guilty or not guilty. If the criminal is guilty, he is guilty. If he is not guilty then it doesn’t mean that he is innocent.  This is what we can relate to the null hypothesis. Also, if the criminal is innocent, he is innocent, he is innocent. If he is not innocent doesn’t mean he is guilty. Court of law protects the innocent. It is considered that an innocent person shouldn’t be punished. This is important than the not guilty case. Type 1 error is the same as an innocent person sent to jail and type 2 error is like the person is guilty and set free. It is more important, that type 1 error is more important than type 2.

If the probability of making a type 1 error is determined by “α”, the probability of a type 2 error is “β”. Beta depends on the power of the test (i.e the probability of not committing a type 2 error, which is equal to 1-β).

Type 1 error = Confidence level = 1 – α

Type 2 error = Power of test = 1 – β

 

If α = 5%, that means it is a good hypothesis test so the hypothesis will be 100 – 5 = 95%

Similarly, the Power of the test is the ability to fail to reject.

 

Steps in Hypothesis testing:

  1. Before data collection, develop a clear research problem.
  2. Before data collection, establish both null and alternate hypothesis
  3. Before data collection, determine appropriate statistical test
  4. Before data collection, choose type1 and error rate α. His could be assumed to be 0.05 unless specified.
  5. Before data collection, state the decision rule: When to reject the null hypothesis
  6. Now gather the sample data
  7. Calculate test statistic
  8. State statistical conclusion based on the p-value
  9. Make inference based on conclusions (If p-value is low, the null hypothesis is rejected)
the p-value is probability.
P < α then reject H0
P >= α then fail to reject H0.
There are certain types of Hypothesis tests which, we will look into in the next upcoming sections.
A link that would give more depth of Type 1 and Type 2 Hypothesis analysis is as follows:

Hypothesis testing is further divided into parametric test and non-parametric test:
Parametric test:
In Statistics, a parametric test is a kind of hypothesis test that gives generalizations for generating records regarding the mean of the primary original population.
Non-Parametric test:
The non-parametric test doesn't require any population distribution, which is meant by distinct parameters. It is also a kind of hypothesis test that is not based on the underlying hypothesis. In the case of a non-parametric test, the test is based on the differences in the median. So, this kind of test is called a distribution-free test. The test variables are determined on the nominal or ordinal level. If the independent variables are non-metric, the non-parametric test is usually performed.


Various types of Parametric and non-parametric tests are as follows:



Various types of tests for hypothesis are as follows:



We will have a look at these tests selectively in upcoming sessions. Till then take care and enjoy!





                                              











 

 


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