Đề thi, bài tập trắc nghiệm online Lý thuyết xác suất và thống kê – Đề 1

1. What is the law of large numbers?

A. The average of the results obtained from a large number of trials should be exactly equal to the expected value.

B. As the number of trials of a random experiment increases, the sample mean approaches the population mean.

C. Larger samples always give less variable means.

D. The sum of probabilities in any experiment must equal a large number.

2. In statistics, what is 'variance' a measure of?

A. Central tendency

B. The most frequent value

C. Dispersion or spread of data points around the mean

D. The middle value in a dataset

3. What is the 'critical region' in hypothesis testing?

A. The region where the null hypothesis is always accepted.

B. The set of values of the test statistic for which the null hypothesis is rejected.

C. The region around the sample mean.

D. The region where the p-value is always greater than the significance level.

4. What is the purpose of stratification in sampling?

A. To make the sample size larger.

B. To ensure every individual in the population has an equal chance of being selected.

C. To reduce sampling error by dividing the population into homogeneous subgroups (strata) and then sampling from each stratum.

D. To introduce bias into the sample for specific research purposes.

5. Which type of error occurs in hypothesis testing when you reject the null hypothesis when it is actually true?

A. Type II error

B. Standard error

C. Type I error

D. Sampling error

6. What is the range of possible values for a probability?

A. Any real number

B. 0 to infinity

C. -1 to 1

D. 0 to 1

7. What is the 'null hypothesis' in hypothesis testing?

A. The hypothesis that the researcher is trying to prove.

B. A statement of no effect or no difference, which is tested against the alternative hypothesis.

C. The hypothesis that is always true.

D. The hypothesis based on sample data.

8. What is the relationship between standard deviation and variance?

A. Standard deviation is the square of the variance.

B. Variance is the square root of the standard deviation.

C. Standard deviation is the square root of the variance.

D. They are unrelated measures.

9. In probability theory, what does it mean for two events to be independent?

A. They cannot occur at the same time.

B. The occurrence of one event affects the probability of the other event.

C. The occurrence of one event does not affect the probability of the other event.

D. They are mutually exclusive.

10. In a right-skewed distribution, how are the mean, median, and mode typically ordered?

A. Mean < Median < Mode

B. Mode < Median < Mean

C. Mean = Median = Mode

D. Median < Mode < Mean

11. What is Bayes' Theorem used for?

A. Calculating the probability of independent events.

B. Updating probabilities based on new evidence.

C. Finding the mean of a distribution.

D. Determining the variance of a distribution.

12. Which of the following is a discrete random variable?

A. Height of a person

B. Temperature of a room

C. Number of cars passing a point on a highway in an hour

D. Time taken to complete a race

13. What is the role of random sampling in statistics?

A. To ensure the sample is exactly the same as the population.

B. To introduce bias into the sample.

C. To minimize bias and ensure the sample is representative of the population, allowing for valid inferences.

D. To make the sample smaller and easier to manage.

14. Which of the following is a measure of dispersion?

A. Median

B. Mode

C. Range

D. Mean

15. What is the purpose of confidence intervals?

A. To estimate the exact population parameter.

B. To provide a range of plausible values for a population parameter based on sample data.

C. To prove the sample statistic is accurate.

D. To calculate the sample mean.

16. What is regression analysis primarily used for?

A. To summarize data

B. To test hypotheses about population means

C. To model the relationship between variables and predict outcomes

D. To measure the spread of data

17. Which probability distribution is best suited for modeling the number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event?

A. Binomial distribution

B. Normal distribution

C. Poisson distribution

D. Exponential distribution

18. For a standard normal distribution, what is the mean and standard deviation?

A. Mean = 1, Standard Deviation = 1

B. Mean = 0, Standard Deviation = 1

C. Mean = 0, Standard Deviation = 0

D. Mean = 1, Standard Deviation = 0

19. What does a correlation coefficient of -1 indicate?

A. No linear relationship between the variables.

B. A perfect positive linear relationship.

C. A perfect negative linear relationship.

D. A strong non-linear relationship.

20. If two events are mutually exclusive, what is the probability of their intersection (both occurring at the same time)?

A. P(A) + P(B)

B. P(A) * P(B)

C. 0

D. 1

21. What does a p-value in hypothesis testing represent?

A. The probability that the null hypothesis is true.

B. The probability that the alternative hypothesis is true.

C. The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

D. The significance level of the test.

22. What is the Central Limit Theorem?

A. The theorem stating that the sample mean always equals the population mean.

B. The theorem stating that the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

C. The theorem stating that the population must be normally distributed.

D. The theorem describing the center of any dataset.

23. Which of the following distributions is continuous?

A. Binomial distribution

B. Poisson distribution

C. Normal distribution

D. Bernoulli distribution

24. In statistics, what is 'degrees of freedom' often related to?

A. The total number of data points.

B. The number of independent pieces of information available to estimate a parameter.

C. The level of significance in hypothesis testing.

D. The range of the data.

25. What does correlation measure?

A. Causation between two variables

B. The strength and direction of a linear relationship between two variables

C. The difference between means of two groups

D. The variability within a single variable

26. What is the purpose of hypothesis testing in statistics?

A. To prove the null hypothesis is true.

B. To calculate the exact population parameter.

C. To provide evidence for or against the null hypothesis based on sample data.

D. To describe the sample data.

27. What is the difference between a population and a sample?

A. A population is always larger than a sample, but a sample is more accurate.

B. A population is the entire group of individuals we are interested in studying, while a sample is a subset of the population that we actually examine.

C. A sample is used in descriptive statistics, and a population is used in inferential statistics.

D. There is no difference; the terms are interchangeable.

28. Which of the following is NOT a measure of central tendency?

A. Mean

B. Median

C. Mode

D. Standard Deviation

29. In the context of sampling distributions, what is the 'standard error'?

A. The error made when calculating the standard deviation of the population.

B. The standard deviation of the sampling distribution of a statistic (like the sample mean).

C. The difference between the sample mean and the population mean.

D. The error due to measurement inaccuracies.

30. What is the fundamental difference between descriptive statistics and inferential statistics?

A. Descriptive statistics describes a sample, while inferential statistics describes a population.

B. Descriptive statistics uses graphs, while inferential statistics uses calculations.

C. Descriptive statistics summarizes data, while inferential statistics makes predictions or generalizations about a larger group based on sample data.

D. Descriptive statistics is more accurate than inferential statistics.