Testing of Hypothesis with statistical formula for test statistics
Table of Contents:
- Introduction
- Hypothesis Testing
- Parametric Tests a. Z-test b. T-test c. ANOVA
- Non-parametric Tests
- Conclusion
Introduction:
Hypothesis testing is an important tool in statistics. It is used to determine whether a certain hypothesis is true or false. This is done by examining the statistical data and using the appropriate formula for test statistics. Hypothesis testing can be classified into two types: Parametric tests and Non-parametric tests.
Parametric Tests:
Parametric tests are used when the underlying distribution of the data is known. They are based on certain assumptions about the data, such as normality or homoscedasticity. The following parametric tests are commonly used:
a. Z-test:
The Z-test is used when the population mean and standard deviation are known. It is used to test the hypothesis that the sample mean is significantly different from the population mean. The formula for the Z-test is:
Z = (x - μ) / (σ / √n)where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
b. T-test:
The T-test is used when the population standard deviation is unknown. It is used to test the hypothesis that the sample mean is significantly different from the population mean. The formula for the T-test is:
t = (x - μ) / (s / √n)where x is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
c. ANOVA:
The ANOVA (Analysis of Variance) test is used when there are more than two groups with different means. It is used to test the hypothesis that the means of the groups are equal. The formula for the ANOVA test is:
F = MSB / MSEwhere MSB is the mean square between groups and MSE is the mean square error.
Non-parametric Tests:
Non-parametric tests are used when the underlying distribution of the data is unknown. They are not based on any assumptions about the data. The following non-parametric tests are commonly used:
a. Wilcoxon rank-sum test:
The Wilcoxon rank-sum test is used to compare two independent samples. It is used to test the hypothesis that the medians of the two samples are equal. The formula for the Wilcoxon rank-sum test is:
U = N1N2 + (N1(N1 + 1))/2 - R1where N1 and N2 are the sample sizes, and R1 is the sum of the ranks of the first sample.
b. Kruskal-Wallis test:
The Kruskal-Wallis test is used to compare three or more independent samples. It is used to test the hypothesis that the medians of the samples are equal. The formula for the Kruskal-Wallis test is:
H = 12 / (n(n + 1))∑(Rj - (n + 1)/2)^2 / τwhere n is the total sample size, Rj is the sum of ranks of the j-th sample, and τ is the number of groups.
Conclusion:
In conclusion, hypothesis testing is an important tool in statistics. It is used to determine whether a certain hypothesis is true or false. Parametric tests are used when the underlying distribution of the data is known, while non-parametric tests are used when the underlying distribution of the data is unknown. Some commonly used parametric tests are Z-test, T-test, and ANOVA. Some commonly used non-parametric tests are Wilcoxon rank-sum test and Kruskal-Wallis test.
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