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Hypothesis Testing for Single Samples: Definition & Examples

Instructor: Prashant Mishra

Prashant is currently pursuing his bachelors in Computer Science and Engineering.

In this lesson, we will discuss the basics underlying hypothesis testing, including population, samples, level of significance and tests of significance. We will also cover the two statistic tools - t-test and F-test.

Population and Samples

Suppose the population of a city is 10 million, and we want to determine whether the new jam made by company X will be preferred by customers. It is practically impossible for us to do an experiment with 10 million people, so we take some people, say 50, and test whether they like the jam.

The 10 million people in the city are referred to as population, and the 50 people undergoing our testing are referred to as samples. We always have some parameters of the population with us while we generate statistics about the samples.

This forms the basis of whatever follows up next. In this lesson, we will discuss how to draw conclusions for the whole population by looking at facts provided by the samples, a small part of the population.

Tests of Significance

Tests done to understand the difference between sample statistics and population statistics are referred to as tests of significance. These tests help us to decide our conclusion.

Before performing tests of significance, we always form two hypotheses. On the basis of the results obtained from the test of significance, we decide which hypothesis should be accepted and which should be rejected.

We create two types of hypotheses:

  • Null Hypothesis (Ho): According to this hypothesis, there is no significant difference between sample statistics and population statistics.
  • Alternate Hypothesis (H1): According to this hypothesis, there is a significant difference between sample statistics and population statistics.

Level of Significance

The probability that a random value of the statistic belongs to the critical region is known as level of significance and is denoted by α. Generally, α is calculated at 5% unless stated otherwise. Critical region is a region associated with a statistic amounting to rejection of a null hypothesis.

If the value of test of significance is less than α, then we accept the null hypothesis and reject the alternate hypothesis. If the value is greater than α, then we reject the null hypothesis and accept the alternate hypothesis.

Student's T-Test

The student's t-test is one of the most famous tests of significance, and it is used when the mean of population and samples are known. The student's t-test is given by:


testt


Let x be the mean of sample, μ be the mean of population, s be standard deviation of the sample, and n be the sample size (number of entities in the sample). The degrees of freedom is given by n - 1 and is used to determine the value of α from a t-table.

Example of Student's T-Test

The mean weekly sales of energy bars in department stores was 146.3 bars per store. After an advertising campaign, the mean weekly sales in 22 stores for a typical week increased to 153.7 and showed a standard deviation of 17.2. Was the advertising campaign successful?

Given n = 22, x = 153.7, μ = 146.3 and s =17.2, we define null and alternate hypothesis as:

  • Ho: μ = 146.3, i.e., the advertising campaign was not successful.
  • H1: μ > 146.3, i.e., the advertising campaign was successful.


t1test


We get t = 1.97.

From t-table, at 5% level of significance with degree of freedom 21, we get α = 1.72. Since t > α, we reject the null hypothesis and accept the alternate hypothesis. Hence, we can conclude that the advertising campaign was successful.

F-Test

The F-test is used when variances of two different samples are known. Let Sx and Sy be variances of two populations. The F-test is given by:


testf


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