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The Behrens–Fisher problem occurs when testing the equality of means of two normal distributions without the assumption that the two variances are equal. This paper presents approaches based on the exact and near-exact distributions for the test statistic of the Behrens–Fisher problem, depending on different combinations of even or odd sample sizes. We present the exact distribution when both sample sizes are odd and the near-exact distribution when one or both sample sizes are even. The near-exact distributions are based on a finite mixture of generalized integer gamma (GIG) distributions, used as an approximation to the exact distribution, which consists of an infinite series. The proposed tests, based on the exact and the near-exact distributions, are compared with Welch’s t-test through Monte Carlo simulations, in particular for small and unbalanced sample sizes. The results show that the proposed approaches are competent solutions to the Behrens–Fisher problem, exhibiting precise sizes and better powers than Welch’s approach for those cases. Numerical studies show that the Welch’s t-test tends to be a bit more conservative than the test statistics based on the exact or near-exact distribution, in particular when sample sizes are small and unbalanced, situations in which the proposed exact or near-exact distributions obtain higher powers than Welch’s t-test.

In this study, we analyzed the changes in the average daily, monthly, seasonal, and annual surface air temperatures based on the temperature data obtained from seven stations (1 January 2005–31 December 2019; 15 years) belonging to the central Baltic area (Stockholm, Tallinn, Helsinki, Narva, Pärnu, Tartu, and Võru). The statistical analysis revealed that there was a strong correlation between the daily average surface air temperature of the studied cities (range: 0.95–0.99). We analyzed the frequency distribution of the average surface air temperatures in addition to the Kruskal–Wallis and Dunn’s tests (significance level of 0.05) to demonstrate that the difference in air temperatures between Narva, Tallinn, Tartu, and Stockholm are critical. The Welch’s t-test (significance level 0.05), used to study the differences in the average monthly air temperature of the cities in question, showed that the surface air temperatures in Stockholm do not differ from Tallinn air temperatures from May to August. However, the surface air temperatures of Narva were similar to those of Tallinn in September. According to our results, the trends in the changes of monthly average surface air temperatures have a certain course during the year (ranging from 1.8 °C (Stockholm) to 4.5 °C (Võru and Tartu) per decade in February). During the entire study period, in addition to February, the surface air temperature increased in all the studied cities in March, May, June, and December, and the surface air temperature did not increase in January or from July to October. During the study period, the average annual surface air temperature in the cities of the central Baltic area increased by 0.43 °C per decade. The results also confirm that the surface air temperature in the study area is changing differently in different cities. The acceleration of the surface air temperature is very alarming and requires a significant intensification of the measures taken to slow down the temperature rise.

The peculiarities of the long-term change of the annual and monthly average air temperatures until 2017 in five cities of the coastal area of the Central Baltic region—Stockholm, Tallinn, Riga, Helsinki, and Saint Petersburg—were studied. The anomalies of the annual and monthly average air temperatures in relation to the average characteristics 1961⁻1990 were analyzed. The trends in the air temperature changes during 1980⁻2017, which come to 0.5 °C per ten years, have been found in the cities of the Central Baltic coastal area. The average air temperature in the Central Baltic cities has grown faster than the global and northern hemisphere. For the longer period of 1850⁻2017, the average annual rise of air temperature was within the range of 0.1 °C per ten years. The rise in temperature in different months is different, and the rise of the of the average temperature in the summer period has not occurred (at a significance level of 0.05). With the analysis of the frequency distributions of the average annual air temperatures and Welch’s t-test, it is demonstrated that the air temperature (at a significance level of 0.05) has risen in all the months only in Saint Petersburg during 1901⁻2017 in comparison to the 19th century. There has been no reliable rise of the air temperature during the century in February and from June to September in Riga, from June to October in Helsinki, from June to September in Stockholm, and in August and September in Tallinn. It was found that the average air temperature trends have a certain annual course. The air temperature has risen most in March and April, reaching 0.09 °C (Stockholm, Tallinn) up to 0.23 °C (Saint Petersburg) per ten years. From June to September, the rise of air temperature is considerably lower, remaining below 0.04 °C per ten years. The changes in air temperature are small during the summer and mid-winter; the air temperature has significantly risen in autumn and spring.

The continued search for sustainable and eco-friendly materials has led to the integration of bio-fibers, such as flax fiber, as reinforcement in composite materials; however, a wide variation in their diameters and mechanical properties poses a considerable challenge for their incorporation in load bearing and structural bio-composite materials. In this paper, a rigorous experimental investigation was performed using two varieties of linseed flax from two growing locations to determine if the variations observed in ultimate tensile strength, Young’s modulus, failure strain and diameter could be attributed to the diameters of the stems that produced the fibers. Tests were performed in two different facilities and the results were compared and analyzed using Welch’s t-tests. Results showed that samples which differed by stem diameter had statistically significant positive correlation with fiber diameter and negative correlation with tensile strength. No correlations for tensile strength, Young’s modulus or fiber diameter were found in samples with the same stem diameter range that were grown in different locations or were of different varieties, that is the effect of location and variety is not statistically significant. Failure strain did not show any statistical significance with respect to differences in stem diameter and only showed one statistically significant result between both facilities for one of the two growing location comparisons.