{"id":6026,"date":"2020-04-24T15:45:00","date_gmt":"2020-04-24T10:15:00","guid":{"rendered":"https:\/\/www.mygreatlearning.com\/blog\/hypothesis-testing-in-r-with-examples-and-case-study\/"},"modified":"2024-11-08T11:23:19","modified_gmt":"2024-11-08T05:53:19","slug":"hypothesis-testing-in-r-with-examples-and-case-study","status":"publish","type":"post","link":"https:\/\/www.mygreatlearning.com\/blog\/hypothesis-testing-in-r-with-examples-and-case-study\/","title":{"rendered":"Hypothesis Testing in R- Introduction Examples and Case Study"},"content":{"rendered":"\n

- By Dr. Masood H. Siddiqui, Professor & Dean (Research) at Jaipuria Institute of Management, Lucknow<\/em><\/p>\n\n\n\n

Introduction to Hypothesis Testing in R<\/strong><\/h2>\n\n\n\n

The premise of Data Analytics is based on the philosophy of the \u201cData-Driven Decision Making<\/strong>\u201d that univocally states that decision-making based on data has less probability of error than those based on subjective judgement and gut-feeling. So, we require data to make decisions and to answer the business\/functional questions. Data may be collected from each and every unit\/person, connected with the problem-situation (totality related to the situation). This is known as Census<\/strong> or Complete Enumeration<\/strong> and the \u2018totality\u2019 is known as Population<\/strong>. Obv.iously, this will generally give the most optimum results with maximum correctness but this may not be always possible. Actually, it is rare to have access to information from all the members connected with the situation. So, due to practical considerations, we take up a representative subset<\/em> from the population, known as Sample<\/strong>. A sample is a representative<\/em> in the sense that it is expected to exhibit the properties of the population, from where it has been drawn. <\/p>\n\n\n\n

So, we have evidence (data) from the sample and we need to decide for the population on the basis of that data from the sample i.e. inferring<\/em> about the population on the basis of a sample. This concept is known as Statistical Inference<\/strong>. <\/p>\n\n\n\n

Before going into details, we should be clear about certain terms and concepts that will be useful:<\/p>\n\n\n\n

Parameter and Statistic<\/strong><\/h3>\n\n\n\n

Parameters<\/strong> are unknown constants<\/em> that effectively define<\/em> the population distribution<\/em>, and in turn, the population<\/em>, e.g. population mean (\u00b5), population standard deviation (\u03c3), population proportion (P) etc. Statistics<\/strong> are the values characterising the sample<\/em> i.e. characteristics of the sample. They are actually functions of sample values<\/em> e. g. sample mean (x\u0304), sample standard deviation (s), sample proportion (p) etc. <\/p>\n\n\n\n

Sampling Distribution<\/strong><\/h3>\n\n\n\n

A large number of samples may be drawn from a population. Each sample may provide a value of sample statistic, so there will be a distribution of sample statistic value<\/em> from all the possible samples i.e. frequency distribution of sample statistic<\/em>. This is better known as Sampling distribution of the sample statistic<\/strong>. Alternatively, the sample statistic is a random variable<\/em>, being a function of sample values (which are random variables themselves). The probability distribution of the sample statistic<\/em> is known as sampling distribution of sample statistic. Just like any other distribution, sampling distribution may partially be described by its mean<\/em> and standard deviation<\/em>. The standard deviation of sampling distribution of a sample statistic<\/em> is better known as the Standard Error<\/strong> of the sample statistic. <\/p>\n\n\n\n

Standard Error<\/strong> <\/h3>\n\n\n\n

It is a measure of the extent of variation<\/em> among different values of statistics from different possible samples. Higher the standard error, higher is the variation among different possible values of statistics. Hence, less will be the confidence that we may place on the value of the statistic for estimation purposes. Hence, the sample statistic having a lower value of standard error is supposed to be better for estimation of the population parameter. <\/p>\n\n\n\n

Example:<\/span><\/h4>\n\n\n\n

1(a).<\/strong> A sample of size \u2018n\u2019 has been drawn for a normal population N (\u00b5, \u03c3). We are considering sample mean (x\u0304) as the sample statistic. Then, the sampling distribution of sample statistic x\u0304 will follow Normal Distribution with mean \u00b5x\u0304<\/sub> = \u00b5 and standard error \u03c3x\u0304<\/sub> = \u03c3\/\u221a<\/strong>n.<\/p>\n\n\n\n

Even if the population is not following the Normal Distribution but for a large sample (n = large), the sampling distribution of x\u0304 will approach to (approximated by) normal distribution with mean \u00b5x\u0304<\/sub> = \u00b5 and standard error \u03c3x\u0304<\/sub> = \u03c3\/\u221a<\/strong>n, as per the Central Limit Theorem<\/a><\/em>. <\/p>\n\n\n\n

(b).<\/strong> A sample of size \u2018n\u2019 has been drawn for a normal population N (\u00b5, \u03c3), but population standard deviation \u03c3 is unknown, so in this case \u03c3 will be estimated by sample standard deviation(s). Then, sampling distribution of sample statistic x\u0304 will follow the student's t distribution (with degree of freedom = n-1) having mean \u00b5x\u0304<\/sub> = \u00b5 and standard error \u03c3x\u0304<\/sub> = s\/\u221a<\/strong>n.<\/p>\n\n\n\n

2.<\/strong> When we consider proportions for categorical data. Sampling distribution of sample proportion p =x\/n (where x = Number of success out of a total of n) will follow Normal Distribution with mean \u00b5p<\/sub> = P and standard error \u03c3p<\/sub> = \u221a(<\/strong>PQ\/n), (where Q = 1-P). This is under the condition that n is large such that both np and nq should be minimum 5.<\/p>\n\n\n\n

Statistical Inference<\/strong><\/h3>\n\n\n\n

Statistical Inference encompasses two different but related problems:<\/p>\n\n\n\n

1. Knowing about the population-values on the basis of data from the sample. This is known as the problem of Estimation<\/strong>. This is a common problem in business decision-making because of lack of complete information and uncertainty but by using sample information, the estimate will be based on the concept of data based decision making. Here, the concept of probability is used through sampling distribution to deal with the uncertainty. If sample statistics is used to estimate the population parameter<\/em>, then in that situation that is known as the Estimator; <\/strong>{like <\/strong>sample <\/strong>mean (x\u0304) to estimate population mean \u00b5, sample proportion (p) to estimate population proportion (P) etc.}. A particular value of the estimator<\/em> for a given sample is known as Estimate<\/strong>. For example, if we want to estimate average sales of 1000+ outlets of a retail chain and we have taken a sample of 40 outlets and sample mean (estimator<\/em>) x\u0304 is 40000. Then the estimate will be 40000.<\/p>\n\n\n\n

There are two types of estimation:<\/p>\n\n\n\n