Đề thi, bài tập trắc nghiệm online Lý thuyết xác suất và thống kê – Đề 13

R-squared (coefficient of determination) in regression analysis indicates the goodness of fit of the model. It represents the proportion of the variance in the dependent variable that is explained or predicted by the independent variable(s). A higher R-squared suggests a better fit.

Independent events are those where the occurrence of one event does not affect the probability of the other event occurring. Dependent events are those where the occurrence of one event does influence the probability of the other event.

The Poisson distribution is used to model the number of times an event occurs in an interval of time or space. It's appropriate when events occur randomly and independently at a constant average rate. Examples include number of phone calls per hour, or defects per square meter.

The correlation coefficient, typically Pearson's r, measures the strength and direction of a linear association between two variables. It ranges from -1 to +1, indicating the direction (positive or negative) and strength (closer to |1| is stronger). Correlation does not imply causation.

Probability theory starts with known probabilities and predicts outcomes. Statistics works in reverse: it uses observed outcomes to infer the probabilities that generated those outcomes.

The expected value (or mean) of a random variable is the long-run average value of repetitions of the experiment it represents. It's a weighted average of all possible values, weighted by their probabilities.

Sensitivity (True Positive Rate) is the ability of a test to correctly identify individuals who have the disease (probability of a positive test result given the disease is present). Specificity (True Negative Rate) is the ability of a test to correctly identify individuals who do not have the disease (probability of a negative test result given the disease is absent).

The mean (average) is calculated by summing all values and dividing by the number of values. Outliers, being extreme values, can significantly skew the sum and therefore the mean. The median (middle value) and mode (most frequent value) are less sensitive to outliers.

A discrete probability distribution describes the probability of occurrences of each value of a discrete random variable. The Binomial distribution, which models the number of successes in a fixed number of independent Bernoulli trials, is a discrete distribution. Normal, Exponential, and continuous Uniform are continuous distributions.

Statistical significance means that the observed result is unlikely to be due to random variation alone, assuming the null hypothesis is true. It does not necessarily imply practical importance (option 1) or absolute truth (option 3), and it is not solely determined by sample size (option 4).

ANOVA (Analysis of Variance) is a statistical test used to compare the means of three or more groups. It determines if there are any statistically significant differences between the means of these groups. While t-tests are used for two groups (option 1), ANOVA extends to multiple groups.

Conditional probability concerns the probability of an event occurring given that another event has already occurred. Option 3 directly illustrates this dependency – the probability of rain is conditioned on the cloudiness of the day.

The Central Limit Theorem is a fundamental theorem in statistics stating that, under mild conditions, the distribution of the sample mean of a random sample from any population will be approximately normal for a large sample size. This holds true irrespective of the shape of the population distribution.

Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

A Type I error (False Positive) occurs when we reject a true null hypothesis. A Type II error (False Negative) occurs when we fail to reject a false null hypothesis. Options 1, 3, and 4 are incorrect definitions or irrelevant distinctions.

Measures of variability describe the spread or dispersion of data. Range (the difference between the maximum and minimum values) is a measure of variability. Mean, median, and mode are measures of central tendency.

Multicollinearity occurs in regression analysis when two or more independent variables are highly correlated with each other. This can cause problems in regression models, making it difficult to interpret the individual effects of predictors and inflate standard errors.

Non-parametric tests are statistical tests that do not rely on assumptions about the shape or parameters of the underlying population distribution. The Chi-square test is a non-parametric test, often used for categorical data. T-tests, ANOVA, and Regression analysis are typically parametric, assuming normality or other specific distributions.

A random variable is a variable whose possible values are numerical outcomes of a random experiment. Its value is not fixed but depends on chance.

The p-value is the probability of observing results as extreme as, or more extreme than, those obtained if the null hypothesis were true. A small p-value (typically less than the significance level, α) suggests strong evidence against the null hypothesis.

A confidence interval is a range of values, derived from sample data, that is estimated to contain the true value of a population parameter with a certain level of confidence. It doesn't give the exact population parameter but a plausible range.

The population is the entire group that we are interested in studying or drawing conclusions about. A sample is a subset of the population that is actually observed and analyzed. We use samples to make inferences about populations.

The law of large numbers states that as the number of trials of a random experiment increases, the average of the results gets closer to the expected value. It's a fundamental principle in probability and statistics.

Stratified sampling is a method used to ensure representation of different subgroups (strata) within a population. The population is divided into strata, and then a random sample is taken from each stratum, often proportionally to the stratum's size in the population.

Nominal data is a type of categorical data where categories are mutually exclusive and unordered. Colors of cars are categories without inherent order, making it nominal. Temperature and height are numerical (ratio or interval). Ranking is ordinal (ordered categorical).

Bootstrapping is a resampling technique used to estimate the sampling distribution of a statistic. It works by repeatedly drawing samples with replacement from the original dataset and recalculating the statistic of interest for each resampled dataset. This allows for estimation of standard errors, confidence intervals, and hypothesis testing, especially when analytical solutions are complex or unavailable.

The null hypothesis (H0) in hypothesis testing is a statement of no effect or no difference. It's the hypothesis we attempt to reject in favor of the alternative hypothesis. It's a starting assumption, not what we expect to find or aim to prove directly.

Measures of dispersion describe the spread of data. Variance, standard deviation, and interquartile range (IQR) all measure dispersion. Median is a measure of central tendency, representing the middle value of a dataset, not its spread.

The Exponential distribution is commonly used to model the time until a specific event occurs in a Poisson process (where events occur continuously and independently at a constant average rate). It's used for modeling waiting times, system reliability, etc.

Degrees of freedom (df) are used in statistical tests to account for the number of restrictions placed on the data due to parameter estimation. They influence the shape of distributions like t-distribution and chi-square distribution, which are used to calculate p-values and critical values.