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Kairat Mynbaev wrote a new post

Tue, 19 May 2026 06:32:36 +0000

Unit 14.2. Excel functions for critical values Chapter 14. Excel statistical functions Unit 14.2. Excel functions for critical values In the previous unit each function takes the argument $latex x$ […]

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Tue, 19 May 2026 06:31:28 +0000

Unit 14.1. Excel functions for distributions Chapter 14. Excel statistical functions Unit 14.1. Excel functions for distributions Here I show how to replace Tables 1 through 9 from the appendix in […]

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Tue, 19 May 2026 06:29:33 +0000

Unit 13.6. Questions for repetitionChapter 13. Bridge to econometrics Unit 13.6. Questions for repetition Using Example 13.2 (UoL exam 2012, zone B) as an illustration, explain why an […]

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Tue, 19 May 2026 06:28:09 +0000

Unit 13.5. Criterion of linear dependence and the dummy variable trapChapter 13. Bridge to econometrics Unit 13.5. Criterion of linear dependence and the dummy variable trap We conclude this chapter with two applications […]

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Tue, 19 May 2026 06:26:31 +0000

Unit 13.4. Linear dependence of vectorsChapter 13. Bridge to econometrics Unit 13.4. Linear dependence of vectors The whole section 3.4 from the book is vague because the true explanation of […]

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Tue, 19 May 2026 06:19:39 +0000

Unit 13.3. Model with two regressorsChapter 13. Bridge to econometrics Unit 13.3. Model with two regressors The purpose is to derive equation (3.11) from the book. This will be done in […]

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Tue, 19 May 2026 06:16:11 +0000

Unit 13.2. How to read Dougherty's bookChapter 13. Bridge to econometrics Unit 13.2. How to read Dougherty’s book For each property use a name, like additivity, homogeneity etc. Divide all […]

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Tue, 19 May 2026 06:14:33 +0000

Unit 13.1. Problems with LSE econometricsChapter 13. Bridge to econometrics Unit 13.1. Problems with LSE econometrics In the exposition below we refer to Dougherty, C. Introduction to […]

Kairat Mynbaev wrote a new post

Tue, 19 May 2026 06:11:38 +0000

Unit 12.4. Two-way ANOVA: decompositions and F-testsChapter 12. Analysis of variance (ANOVA) Unit 12.4. Two-way ANOVA: decompositions and F-tests Leading idea. From Eq. (12.6) it follows that $latex bar […]

Kairat Mynbaev wrote a new post

Tue, 19 May 2026 06:10:00 +0000

Unit 12.3. Two-way ANOVA: setup and population model Chapter 12. Analysis of variance (ANOVA) Unit 12.3. Two-way ANOVA: setup and population model Difference between one-way and two-way ANOVA. One-way […]

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Tue, 19 May 2026 06:08:31 +0000

Unit 12.2. One-way ANOVA: distributions and the F-testChapter 12. Analysis of variance (ANOVA) Unit 12.2. One-way ANOVA: distributions and the F-test In the next theorem the result on distribution of […]

Kairat Mynbaev wrote a new post

Tue, 19 May 2026 06:05:29 +0000

Unit 12.1. One-way ANOVA: setup and the SST decomposition Chapter 12. Analysis of variance (ANOVA) Unit 12.1. One-way ANOVA: setup and the $latex SST$ decomposition ANOVA comes with several complex formal […]

Kairat Mynbaev wrote a new post

Tue, 19 May 2026 06:03:15 +0000

Unit 11.4. Questions for repetitionChapter 11. Maximum likelihood method Unit 11.4. Questions for repetition $latex quad$1. Using a sample on the horizontal axis and several candidate […]

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Tue, 19 May 2026 06:02:00 +0000

Unit 11.3. Application to the linear modelChapter 11. Maximum likelihood method Unit 11.3. Application to the linear model Exercise 11.3. Suppose observations are generated according to the […]

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Tue, 19 May 2026 06:00:21 +0000

Unit 11.2. Exponential and Bernoulli distributionsChapter 11. Maximum likelihood method Unit 11.2. Exponential and Bernoulli distributions Exercise 11.1. Suppose the observations come from an […]

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Tue, 19 May 2026 05:57:54 +0000

Unit 11.1. Main idea and algorithm Chapter 11. Maximum likelihood method Unit 11.1. Main idea and algorithm The main idea of the maximum likelihood (ML) method is illustrated in […]

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Mon, 18 May 2026 16:13:46 +0000

Unit 10.6. Questions for repetitionChapter 10. Simple regression Unit 10.6. Questions for repetition Explain the setup of simple regression: what is observed, what is the assumed […]

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Mon, 18 May 2026 16:12:09 +0000

Unit 10.5. Goodness of fitChapter 10. Simple regression Unit 10.5. Goodness of fit Definition 10.7 (Coefficient of determination). The coefficient of determination is defined by […]

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Mon, 18 May 2026 16:08:44 +0000

Unit 10.4. Orthogonality and its consequencesChapter 10. Simple regression Unit 10.4. Orthogonality and its consequences Definition 10.4 (Fitted value and residuals). After having estimated the […]

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Mon, 18 May 2026 15:57:12 +0000

Unit 10.3.5. Statistics for testing hypothesesChapter 10. Simple regression Unit 10.3 Section 10.3.5. Statistics for testing hypotheses Definition 10.3 (Standard errors of OLS estimators). […]

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Mon, 18 May 2026 15:48:03 +0000

Unit 10.3.4. Variances of OLS estimatorsChapter 10. Simple regression Unit 10.3. Section 10.3.4. Variances of OLS estimators In addition to the basic assumptions of Unit 10.2 suppose […]

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Mon, 18 May 2026 15:45:05 +0000

Unit 10.3. Unbiasedness of OLS estimatorsChapter 10. Simple regression Unit 10.3. Ordinary Least Squares (OLS): definition and derivation Section 10.3.3. Unbiasedness of OLS estimators […]

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Mon, 18 May 2026 15:11:48 +0000

Unit 10.3. Ordinary Least Squares (OLS): definition and derivation Chapter 10. Simple regression Unit 10.3. Ordinary Least Squares (OLS): definition and derivation Section 10.3.1. Definition 10.1. OLS estimators […]

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Mon, 18 May 2026 14:56:15 +0000

Unit 10.2. Linear model setupChapter 10. Simple regression Unit 10.2. Linear model setup We assume that vectors $latex (x_1, y_1), ldots, (x_n, y_n)$ are observed and want to […]

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Mon, 18 May 2026 14:54:36 +0000

Unit 10.1. Algebra of sample meansChapter 10. Simple regression Unit 10.1. Algebra of sample means This chapter will be densely populated. It’s because solutions to some exercises from […]

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Mon, 18 May 2026 14:53:17 +0000

Unit 9.7. Questions for repetitionChapter 9. Hypothesis Testing Unit 9.7. Questions for repetition $latex quad$1. In the context of a university admission test discuss the trade-off […]

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Mon, 18 May 2026 14:52:00 +0000

Unit 9.6. Testing for equality of two sample variances Chapter 9. Hypothesis Testing Unit 9.6. Testing for equality of two sample variances Definition 9.6 ($latex F$-distribution). Let $latex chi_m^2$ and […]

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Mon, 18 May 2026 14:50:38 +0000

Unit 9.5. Power of a test Chapter 9. Hypothesis Testing Unit 9.5. Power of a test There are not many exercises for assessing the power of a test, so this will probably be your […]

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Mon, 18 May 2026 14:48:28 +0000

Unit 9.4. p-values Chapter 9. Hypothesis Testing Unit 9.4. $latex p$-values Preamble. Consider the situation in Figure 9.4. Let us think about the dynamics of the […]

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Mon, 18 May 2026 14:47:12 +0000

Unit 9.3. Confidence intervals and hypothesis testing Chapter 9. Hypothesis Testing Unit 9.3. Confidence intervals and hypothesis testing Exercise 9.2. A cereal package should weigh $latex w_0$. If it […]

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Mon, 18 May 2026 14:45:45 +0000

Unit 9.2. Tradeoff between Type I and Type II errors Chapter 9. Hypothesis Testing Unit 9.2. Tradeoff between Type I and Type II errors We always try to minimize the probability of error and maximize the […]

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Mon, 18 May 2026 14:44:12 +0000

Unit 9.1. Concepts of hypothesis testingChapter 9. Hypothesis Testing Unit 9.1. Concepts of hypothesis testing Definition 9.1 (Hypotheses). In hypothesis testing we assume two possibilities, […]

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Mon, 18 May 2026 10:40:40 +0000

Unit 8.5. Questions for repetitionChapter 8. Estimation: One or Two Populations Unit 8.5. Questions for repetition Ex. 1. Show that existence of one unbiased estimator of some parameter […]

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Mon, 18 May 2026 10:39:28 +0000

Unit 8.4.5. Difference between two population proportionsChapter 8. Estimation: One or Two Populations Unit 8.4.5. Difference between two population proportions Exercise 8.9. Suppose that a random sample […]

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Mon, 18 May 2026 10:37:40 +0000

Unit 8.4.4. Independent samples. Case III (Satterthwaite's formula)Chapter 8. Estimation: One or Two Populations Unit 8.4.4. Independent samples. Case III (Satterthwaite’s formula) Assumptions. There are two samples of […]

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Mon, 18 May 2026 10:36:13 +0000

Unit 8.4.3. Independent samples. Case IIChapter 8. Estimation: One or Two Populations Unit 8.4.3. Independent samples. Case II When $latex sigma_X^2 = sigma_Y^2 = sigma^2$ from Eq. (8.8) […]

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Mon, 18 May 2026 10:34:42 +0000

Unit 8.4.2. Independent samples. Case IChapter 8. Estimation: One or Two Populations Unit 8.4.2. Independent samples. Case I Example 8.4. When efficiency of a new medicine is tested, […]

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Mon, 18 May 2026 10:33:01 +0000

Unit 8.4.1. Matched pairs Chapter 8. Estimation: One or Two Populations Unit 8.4.1. Matched pairs As we move into a more applied area, problem statements, their solutions and […]

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Mon, 18 May 2026 09:49:02 +0000

Unit 8.3.3. Confidence interval for population proportionChapter 8. Estimation: One or Two Populations Unit 8.3.3. Confidence interval for population proportion From Eq. (5.14) we know that $latex Ehat p = […]

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Mon, 18 May 2026 09:47:46 +0000

Unit 8.3.2. Confidence interval for the population mean. Case of an unknown variance Chapter 8. Estimation: One or Two Populations Unit 8.3.2. Confidence interval for the population mean. Case of an unknown variance Assumptions. A […]

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Mon, 18 May 2026 09:12:31 +0000

Unit 8.3.1. Confidence interval for the population mean. Case of a known varianceChapter 8. Estimation: One or Two Populations Unit 8.3.1. Confidence interval for the population mean. Case of a known variance Assumptions. A sample […]

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Mon, 18 May 2026 09:11:00 +0000

Unit 8.2. Consistency Chapter 8. Estimation: One or Two Populations Unit 8.2. Consistency This is one of the topics most students don’t understand, partly because too little […]

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Mon, 18 May 2026 09:09:29 +0000

Unit 8.1. Unbiasedness and efficiencyChapter 8. Estimation: One or Two Populations Unit 8.1. Unbiasedness and efficiency Exercise 8.1. If there exists one unbiased estimator of a […]

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Mon, 18 May 2026 08:45:50 +0000

Unit 7.4. Questions for repetitionChapter 7. Sampling and Sampling Distribution Unit 7.4. Questions for repetition Ex. 1. Fill out the next table: Population formula […]

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Mon, 18 May 2026 08:26:18 +0000

Unit 7.3. Monte Carlo simulationsChapter 7. Sampling and Sampling Distribution Unit 7.3. Monte Carlo simulations Here we explain how the Bernoulli, binomial and normal variables can be […]

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Mon, 18 May 2026 08:24:28 +0000

Unit 7.2. Sampling distribution of sample variance Chapter 7. Sampling and Sampling Distribution Unit 7.2. Sampling distribution of sample variance Definition 7.4 (Unbiased and biased estimators). Let […]

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Mon, 18 May 2026 08:23:01 +0000

Unit 7.1. Sampling distributionChapter 7. Sampling and Sampling Distribution Unit 7.1. Sampling distribution Example 7.1. Suppose we are interested in finding the average income […]

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Mon, 18 May 2026 08:21:38 +0000

Unit 6.10. Questions for repetitionChapter 6. Continuous Random Variables and Probability Distributions Unit 6.10. Questions for repetition Discuss the five properties of integrals that […]

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Mon, 18 May 2026 08:11:51 +0000

Unit 6.9. Exponential distribution and link to PoissonChapter 6. Continuous Random Variables and Probability Distributions Unit 6.9. Exponential distribution and link to Poisson Motivating example. You are […]

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Mon, 18 May 2026 08:09:55 +0000

Unit 6.8. Joint distribution and independenceChapter 6. Continuous Random Variables and Probability Distributions Unit 6.8. Joint distribution and independence All definitions in this section have […]