Inferential statistics

Course description: The Inferential Statistics course provides an introduction to the methods used to draw conclusions from data. It focuses on how information from samples can be used to make reliable inferences about larger populations. Students study probability theory, sampling distributions, estimation methods, and hypothesis testing. The course explains key concepts such as confidence intervals, statistical significance, and hypothesis tests. Practical examples and exercises demonstrate how inferential methods are applied in scientific and social research. Attention is also given to interpreting statistical results correctly and understanding the assumptions behind common techniques. The material combines theoretical foundations with analytical and problem-solving skills. By the end of the course, students are able to analyze data critically and evaluate empirical evidence. The course supports the development of quantitative reasoning and evidence-based decision making. Overall, it offers a solid foundation in modern inferential statistical methods. 

1. Important distributions used in statistics

The first course introduces the basic concepts of classical probability theory, with particular emphasis on the interpretation of probability and the concept of random variables. Students will learn about the role of cumulative distribution function and probability density function, as well as their relationship in discrete and continuous cases. In addition, the significance of expected value, variance, and higher-order (central) moments in characterizing random variables will be discussed. Furthermore, this course will cover the most important discrete and continuous distributions used in statistics, such as the binomial distribution, Poisson distribution, normal distribution, Student's distribution, chi-square distribution, and Fisher-Snedecor distribution. 

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2. The central limit theorem and the law of large numbers

When dealing with a wide variety of random phenomena that occur in reality, we often assume that the probability variables in question are normally distributed. The reason for this is that the sum of a large number of independent random variables is always approximately normally distributed if the fluctuations of the individual variables are small compared to the random fluctuations of the sum, regardless of the probability distribution of the individual variables (see the central limit theorem). This course will cover the central limit theorem, the de Moivre-Laplace theorem for binomial distributions, Markov inequality and Chebyshev inequality, and the weak law of large numbers. 

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3. Effectiveness of statistical estimates

The effectiveness of statistical estimates plays a key role in the reliability of data analysis and conclusions. The expected value of an unbiased estimate is equal to the true value of the estimated parameter, while a consistent estimate gets closer and closer to the true parameter as the sample size increases. An asymptotically unbiased estimate becomes unbiased for large samples, even if it shows bias for small samples. Efficient and unbiased estimates have the smallest possible variance under the given conditions, thus allowing for the most accurate conclusions. This course is about the efficiency of statistical estimates, mentioning unbiased, consistent, asymptotically unbiased, efficient, and unbiased estimates.  

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4. Confidence interval of the expected value and of the standard deviation

The confidence interval of the expected value shows the range within which the true population mean is likely to fall based on a sample, thus helping to assess the accuracy of the estimate. This is particularly important in statistical conclusions because it not only gives a single value but also expresses the degree of uncertainty. The interval can usually be calculated from the sample mean and standard deviation, for example, in the case of a normal distribution, using Student's t-distribution (William Sealy Gosset). The confidence interval for the standard deviation can be determined in a similar way, usually using the chi-square distribution, which takes into account the uncertainty of the variance estimate. In this course, I will use the Lagrange multiplier method to show why we construct a symmetric interval for the expected value and how we approximate the appropriate quantiles using tables and Excel. 

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5. The one-sample z-test for the expected value

This course presents the steps involved in the one-sample z-test. In a one-sample z-test, the statistical hypothesis testing process begins with the formulation of the null hypothesis (H_0) and the alternative hypothesis (H_A), and continues with the specification of the significance level (p). During the test, we calculate the test statistic and determine the critical region; if the test statistic falls within this region, we reject H_0. We commit a type I error if we reject a null hypothesis that is actually true, while in the case of a type II error, we fail to reject a false null hypothesis. We make the decision by comparing the test statistic with the critical value(s). The power of the test (power function) gives the probability that the test correctly rejects the null hypothesis when a given alternative is true.

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6. The one-sample t-test for the expected value

The one-sample t-test is a fundamental method of statistical hypothesis testing used when we wish to compare the expected value of a population with a predetermined value, and the population standard deviation is unknown. The hypothesis testing process begins with the formulation of the null hypothesis and the alternative hypothesis, and continues with the specification of the significance level. During the test, we calculate the test statistic and determine the critical region; if the test statistic falls within this region, we reject the null hypothesis. The test statistic follows a Student’s t-distribution, the shape of which depends on the degrees of freedom. We commit a type I error if we reject a null hypothesis that is actually true, while in the case of a type II error, we fail to reject a false null hypothesis. We make the decision by comparing the test statistic with the critical values. 

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7. The chi-square test for the variance

A statistical method frequently used to test the variance of a population is Pearson’s chi-square test, which is designed to determine whether the variance of the population under study corresponds to a pre-specified value. As the first step of the test, we formulate the null hypothesis and the alternative hypothesis, then set the significance level. Next, based on the sample, we determine the value of the chi-square test statistic and define the critical range that serves as the basis for the decision. If the calculated value falls within this range, we reject the null hypothesis. During hypothesis testing, there is a possibility of making an incorrect decision: in the case of a type I error, we reject a null hypothesis that is actually true, while in the case of a type II error, we accept a false null hypothesis. 

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8. The F-test for variances

The F-test is a statistical method used to compare the variances of two populations, allowing us to determine whether there is a significant difference between the variances of the two populations. During hypothesis testing, we first define the null hypothesis and the alternative hypothesis, then specify the chosen significance level. Based on the samples, we calculate the F-test statistic, which is based on the ratio of the two sample variances and follows the Fisher-Snedecor F-distribution. To make a decision, we determine the critical region and then compare the calculated test statistic with the critical values. If the test statistic falls within the rejection region, we reject the null hypothesis and conclude that the variances of the two populations differ. In the test, a type I error occurs when we reject a true null hypothesis, while a type II error occurs when we fail to reject a false null hypothesis. 

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9. The two-sample z-test for the expected values

The two-sample u-test is a method used to statistically compare the means of two different populations (in this case, the theoretical standard deviations of the two populations are known). The purpose of the test is to determine whether there is a significant difference between the expected values of the two populations. The hypothesis testing process begins with the formulation of the null hypothesis and the alternative hypothesis, and continues with the selection of the significance level. Based on the data calculated from the samples, we determine the value of the test statistic, which we compare to the appropriate critical values. We reject the null hypothesis if the result falls within the rejection region. During the decision-making process, a type I error may occur when we reject a true null hypothesis, or a type II error when we accept a false null hypothesis.   

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10. The two-sample t-test for the expected values

The two-sample t-test is a statistical method used to compare the means of two populations; it is primarily used when the population standard deviations are unknown (although the use of the F-test for variances supports the assumption that the standard deviations are equal). The purpose of the test is to determine whether there is a statistically significant difference between the expected values of the two populations. During hypothesis testing, we first formulate the null hypothesis and the alternative hypothesis, then specify the chosen significance level. Based on the sample data, we calculate the test statistic, which follows a Student’s t-distribution, and then compare it with the corresponding critical values. We reject the null hypothesis if the test statistic falls within the rejection region. In the decision process, a type I error may occur when we reject a true null hypothesis, or a type II error when we fail to reject a false null hypothesis.  

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11. Goodness-of-fit-test

A goodness-of-fit test is a statistical method designed to determine whether the distribution of an observed data set conforms to a pre-specified theoretical distribution. We speak of a pure goodness-of-fit test when the parameter does not need to be estimated, and our hypothesis clearly refers to a single distribution function. The goodness-of-fit test is estimative if our hypothesis specifies only the nature of the distribution, i.e., the type of distribution function (normal, exponential, binomial, etc.), and the parameters of the distribution function must be estimated from the sample. During the test, we first formulate the null hypothesis and the alternative hypothesis, then determine the significance level. Next, based on the observed and theoretically expected frequencies, we calculate the test statistic, which typically follows a chi-square distribution. In making the decision, we compare the calculated value with the critical value, and if the test statistic falls within the rejection region, we reject the null hypothesis.

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12. Test of homogeneity

A homogeneity test is a statistical procedure used to determine whether multiple populations or samples come from the same distribution with respect to a particular characteristic under study. In this test, we first formulate the null hypothesis, which states that the populations under study are homogeneous, and the alternative hypothesis, which assumes a difference between them. In the next step, we determine the significance level, then calculate the test statistic based on the samples, which typically follows a chi-square distribution. To make a decision, we define the critical region and compare the calculated value with the critical value. If the test statistic falls within the rejection region, we reject the null hypothesis, meaning we conclude that there is a significant difference between the samples. In a type I error during the test, we consider a population group that is actually homogeneous to be different, while in a type II error, we fail to recognize the actual differences.  

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13. Test of independence

A test of independence is a statistical procedure used to determine whether a statistical relationship exists between two variables or whether they can be considered independent of each other. During the test, the observed frequencies are compared with the expected frequencies calculated under the assumption of independence, and based on the difference between them, we determine the test statistic, which typically follows a chi-square distribution. Using the resulting value, we decide whether to accept or reject the null hypothesis. If the difference is significant, we conclude that there is a relationship between the two variables. When applying this method, we must take into account the possibility of type I and type II errors, as well as the power of the test, which indicates the probability that the test will detect the actual relationship between the variables.

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