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The null hypothesis is a fundamental concept in statistical testing that posits no effect or relationship between variables, serving as a default or baseline assumption to be tested against. It is typically rejected or not rejected based on the strength of evidence provided by sample data, guiding researchers in making inferences about the population.
The alternative hypothesis is a statement in statistical hypothesis testing that proposes a potential effect or relationship between variables, contrary to the null hypothesis which suggests no effect or relationship exists. It is what researchers aim to support through evidence gathered from data analysis, and its acceptance implies that the observed data is statistically significant.
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The p-value is a statistical measure that helps researchers determine the significance of their results by quantifying the probability of observing data at least as extreme as the actual data, assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis, often guiding decisions on hypothesis rejection in favor of the alternative hypothesis.
A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter with a specified level of confidence. It provides a measure of uncertainty around the estimate, allowing researchers to make inferences about the population with a known level of risk for error.
A Type I Error occurs when a true null hypothesis is incorrectly rejected, often referred to as a 'false positive'. It is controlled by the significance level (alpha), which represents the probability of making this error in hypothesis testing.
A Type II error occurs when a statistical test fails to reject a false null hypothesis, leading to a false negative result. It is inversely related to the power of a test, meaning that as the probability of a Type II error decreases, the test's ability to detect an effect when there is one increases.
The significance level, often denoted by alpha, is the threshold for determining whether a statistical hypothesis test result is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true, and is commonly set at 0.05 or 5%, indicating a 5% risk of concluding that a difference exists when there is no actual difference.
A sampling distribution is the probability distribution of a given statistic based on a random sample, and it reflects how the statistic would behave if we repeatedly sampled from the same population. It is crucial for making inferences about population parameters, as it allows us to understand the variability and reliability of the sample statistic.
The Central Limit Theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size becomes larger, regardless of the population's original distribution. This theorem is foundational in statistics because it allows for the application of inferential techniques to make predictions and decisions based on sample data.
Maximum Likelihood Estimation (MLE) is a statistical method for estimating the parameters of a model by maximizing the likelihood function, thereby making the observed data most probable under the assumed statistical model. It is widely used due to its desirable properties such as consistency, efficiency, and asymptotic normality, which make it a cornerstone of statistical inference and machine learning.
The Likelihood Ratio Test is a statistical method used to compare the goodness of fit between two competing models, typically a null model and an alternative model, by evaluating the ratio of their likelihoods. It is a powerful tool for hypothesis testing, especially in the context of nested models, where one model is a special case of the other.
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ANOVA, or Analysis of Variance, is a statistical method used to determine if there are significant differences between the means of three or more independent groups. It helps in understanding whether the observed variations between group means are due to actual differences or random chance.
The Chi-Square Test is a statistical method used to determine if there is a significant association between categorical variables. It compares the observed frequencies in each category to the frequencies expected under the null hypothesis of no association.
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A t-test is a statistical method used to determine if there is a significant difference between the means of two groups, which may be related in certain features. It is commonly used when the data sets, typically small, follow a normal distribution and have unknown variances.
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A Z-Test is a statistical method used to determine if there is a significant difference between sample and population means or between the means of two samples, assuming the data follows a normal distribution and the population variance is known. It is particularly useful when dealing with large sample sizes, as the Central Limit Theorem ensures the sampling distribution of the mean is approximately normal.
Likelihood estimation is a statistical method used to infer the parameters of a model by maximizing the likelihood function, which measures how well the model explains the observed data. It is foundational in both frequentist and Bayesian statistics, serving as a basis for methods such as Maximum Likelihood estimation (MLE) and Bayesian inference.
Sequential testing is a statistical method that allows for the evaluation of data as it is collected, rather than waiting for all data to be gathered before analysis. This approach enables researchers to make decisions or adjust hypotheses in real-time, potentially reducing the number of observations needed and increasing the efficiency of the study.
Confidence intervals in diagnostic testing provide a range of values within which the true value of a diagnostic parameter, such as sensitivity or specificity, is expected to lie with a certain level of confidence, typically 95%. This statistical tool helps in understanding the precision and reliability of diagnostic tests, allowing healthcare professionals to make informed decisions based on the potential variability of test results.
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