Probability and Statistics Quizzes - Test Your Knowledge

Question 1

When a fair coin is flipped three times, what's the chance of getting at least two heads?

Accepted Answer

3/8

Answer

1/8

Answer

1/2

Answer

1/4

Question 2

For the Central Limit Theorem to be applicable, which condition must be met?

Accepted Answer

The sample size is large enough (usually over 30).

Answer

The population standard deviation is known.

Answer

The population distribution is normal.

Answer

Sampling is done with replacement.

Question 3

Which characteristic is not true for a probability distribution?

Accepted Answer

Probabilities can be negative.

Answer

The sum of probabilities for all outcomes equals 1.

Answer

The mean is a weighted average of the possible outcomes.

Answer

The standard deviation measures the dispersion of the distribution.

Question 4

Which of the following is NOT a probability distribution used in statistical modeling?

Accepted Answer

Median

Answer

Poisson distribution

Answer

Exponential distribution

Answer

Normal distribution

Question 5

Which of the following probability distributions is continuous?

Accepted Answer

Normal

Answer

Bernoulli

Answer

Poisson

Answer

Binomial

Question 6

Which of the following statements about the normal distribution is not true?

Accepted Answer

Its points of inflection are located at one standard deviation from the mean.

Answer

It is a continuous probability distribution.

Answer

Its standard deviation measures the spread of data around the mean.

Answer

Its total area under the curve equals 1.

Question 7

Which of the following is NOT a measure of central tendency in probability and statistics?

Accepted Answer

Range

Answer

Mean

Answer

Median

Answer

Mode

Question 8

A fair six-sided die is rolled three times. What is the probability of rolling a different number each time?

Accepted Answer

1/216

Answer

1/6

Answer

1/12

Answer

1/36

Question 9

When a fair six-sided die is rolled twice, what's the probability that the total of the numbers rolled will be more than 8?

Accepted Answer

5/36

Answer

1/6

Answer

1/2

Answer

1/3

Question 10

In a dataset, the covariance between variables X and Y is -0.5. What does this imply about the relationship between X and Y?

Accepted Answer

X and Y are inversely correlated.

Answer

X and Y have a strong positive correlation.

Answer

X and Y are independent.

Answer

The given information is insufficient to determine the relationship between X and Y.

Question 11

On a scatter plot showing the relationship between variables X and Y, a negative correlation coefficient indicates that:

Accepted Answer

As X increases, Y tends to decrease.

Answer

As X increases, Y tends to increase.

Answer

There is no clear relationship between X and Y.

Answer

X and Y are perfectly related.

Question 12

When the covariance between variables X and Y is positive, what can you conclude about their relationship?

Accepted Answer

X and Y are positively correlated.

Answer

X and Y are negatively correlated.

Answer

X and Y are independent.

Answer

The relationship between X and Y cannot be determined from covariance alone.

Question 13

In probability theory, which approach assumes equally likely outcomes and uses mathematical calculations to determine probabilities?

Accepted Answer

Theoretical approach

Answer

Empirical approach

Answer

Subjective approach

Question 14

Which of the following is not a type of probability distribution?

Accepted Answer

Median

Answer

Binomial distribution

Answer

Normal distribution

Answer

Poisson distribution

Question 15

In a class, 60% of students have brown hair. If a student is chosen at random, what's the chance they have brown hair?

Accepted Answer

60%

Answer

40%

Answer

80%

Answer

100%

Question 16

In a class of 100 students, 30% have blue eyes. If a student is randomly chosen, what's the chance they'll have blue eyes?

Accepted Answer

30%

Answer

70%

Answer

50%

Answer

10%

Question 17

Which of the following is the probability mass function of a Bernoulli random variable with parameter p?

Accepted Answer

P(X = 1) = p, P(X = 0) = 1 - p

Answer

P(X = x) = (1 - p) * x

Answer

P(X = x) = e^(-x) * x^n / n!

Answer

P(X = x) = 1 / n

Question 18

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

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Mode

Question 19

Which of the following is a non-parametric statistical test?

Accepted Answer

Spearman's rank correlation coefficient

Answer

t-test

Answer

ANOVA

Answer

Chi-square test

Question 20

Which of the following is a Type I error?

Accepted Answer

Rejecting the null hypothesis when it is true.

Answer

Accepting the null hypothesis when it is false.

Answer

Failing to reject the null hypothesis when it is false.

Answer

Rejecting the alternative hypothesis when it is true.

Question 21

How is a population distinct from a sample?

Accepted Answer

A population is the entire group of interest, while a sample is a subset of that population.

Answer

A sample is the entire group of interest, while a population is a subset of that sample.

Answer

There is no distinction between a population and a sample.

Answer

A population is a group of individuals, while a sample is a group of objects.

Question 22

In hypothesis testing, what is the null hypothesis?

Accepted Answer

The statement that there is no significant difference between the groups.

Answer

The statement that there is a significant difference between the groups.

Answer

The statement that the probability of an event is 0.5.

Answer

The statement that the sample mean is equal to the population mean.

Question 23

What is the formula for the sample standard deviation?

Accepted Answer

Standard deviation = Σ(x - μ)² / (N - 1)

Answer

Standard deviation = Σ(x - μ)² / N

Answer

Standard deviation = Σx² / N

Answer

Standard deviation = Σx / N

Question 24

What is the purpose of a confidence interval?

Accepted Answer

To estimate the range of values within which the true population parameter is likely to fall.

Answer

To test whether a parameter is equal to a specific value.

Answer

To compare two different parameters.

Answer

To predict the future value of a variable.

Question 25

From a population of 1,000 students, a sample of 100 students is drawn. What is the sampling error in this scenario?

Accepted Answer

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

Answer

The standard deviation of the sample.

Answer

The probability of selecting a male student.

Answer

The count of female students in the sample.

Question 26

A confidence interval is calculated with a 95% confidence level. What is the likelihood that the true population mean falls outside the interval?

Accepted Answer

0.025

Answer

0.05

Answer

0.95

Answer

0.975

Question 27

Which of the following statistical tests is non-parametric?

Accepted Answer

Wilcoxon rank-sum test

Answer

t-test

Answer

ANOVA

Answer

Chi-square test

Question 28

What is the primary purpose of hypothesis testing in statistics?

Accepted Answer

To determine if there is a statistically significant difference between two groups.

Answer

To predict future events.

Answer

To estimate the population mean.

Answer

To describe the distribution of a variable.

Question 29

In a class of 25 students, 15 are male and 10 are female. What's the probability of choosing a male or female student randomly from this class?

Accepted Answer

0.5

Answer

0.25

Answer

0.75

Answer

1

Question 30

Which sampling technique aims to select a representative sample from a population?

Accepted Answer

Simple random sampling

Answer

Stratified sampling

Answer

Systematic sampling

Answer

Cluster sampling

Question 31

In a statistical test, what does the null hypothesis imply?

Accepted Answer

The observed data is purely due to chance.

Answer

The alternative hypothesis is valid.

Answer

The sample mean is equivalent to the population mean.

Answer

The difference between two sample means is negligible.

Question 32

Which statistical test is appropriate for comparing the means of **two independent** groups?

Accepted Answer

t-test

Answer

Chi-square test

Answer

ANOVA

Answer

Regression analysis

Question 33

What does the correlation coefficient represent?

Accepted Answer

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

Answer

The likelihood of two events occurring together.

Answer

The average value of a distribution.

Answer

The spread of a distribution.

Question 34

What is the primary purpose of a probability model?

Accepted Answer

To represent the uncertainty associated with random events.

Answer

To accurately predict future events.

Answer

To analyze patterns in data.

Answer

To guide decision-making processes.

Question 35

In a sequence of independent experiments, the number of successes follows which probability distribution?

Accepted Answer

Binomial distribution

Answer

Poisson distribution

Answer

Normal distribution

Answer

Uniform distribution

Question 36

In hypothesis testing, what is the distinction between a Type I and Type II error?

Accepted Answer

Type I error: rejecting a true null hypothesis; Type II error: failing to reject a false null hypothesis

Answer

Type I error: failing to reject a true null hypothesis; Type II error: rejecting a false null hypothesis

Answer

Type I error: not rejecting a false null hypothesis; Type II error: rejecting a true null hypothesis

Answer

Type I error: rejecting a false null hypothesis; Type II error: not rejecting a true null hypothesis

Question 37

Which measure of spread is highly susceptible to outliers in a dataset?

Accepted Answer

Range

Answer

Variance

Answer

Standard deviation

Answer

Interquartile range

Question 38

Which sampling method involves dividing the population into homogeneous groups (strata) and randomly selecting elements from each stratum?

Accepted Answer

Stratified sampling

Answer

Simple random sampling

Answer

Systematic sampling

Answer

Cluster sampling

Question 39

In a hypothesis test for the equality of two population means, what is the null hypothesis?

Accepted Answer

μ1 = μ2

Answer

μ1 > μ2

Answer

μ1 ≠ μ2

Answer

μ2 > μ1

Question 40

Which assumption is crucial for the validity of the t-test for a single sample mean?

Accepted Answer

The population distribution is normal.

Answer

The population variance is known.

Answer

The sample size is small.

Answer

The sample is randomly selected.

Question 41

What is the primary purpose of a confidence interval?

Accepted Answer

To estimate the true population parameter

Answer

To test the null hypothesis

Answer

To determine the significance of a relationship

Answer

To predict future values

Question 42

Which measure of dispersion represents the average distance of data points from their mean?

Accepted Answer

Standard deviation

Answer

Range

Answer

Variance

Answer

Interquartile range

Question 43

Distinguish between a parameter and a statistic.

Accepted Answer

A parameter is a measure of a population, while a statistic is a measure of a sample.

Answer

A statistic is a measure of a population, while a parameter is a measure of a sample.

Answer

A parameter is a fixed value, while a statistic is a random variable.

Answer

A statistic is a fixed value, while a parameter is a random variable.

Question 44

Which of the following tests is most appropriate to determine if there is a significant difference between the means of two independent populations?

Accepted Answer

Two-sample t-test

Answer

One-sample t-test

Answer

Chi-squared test

Answer

ANOVA

Question 45

Which of the following is a non-parametric test that can be used to compare the medians of two independent populations?

Accepted Answer

Mann-Whitney U test

Answer

Kolmogorov-Smirnov test

Answer

Chi-squared test

Answer

Fisher's exact test

Question 46

Which of the following is **not** a measure of central tendency?

Accepted Answer

Range

Answer

Mean

Answer

Median

Answer

Mode

Question 47

Which of the following is a measure of central tendency?

Accepted Answer

Mean

Answer

Range

Answer

Standard deviation

Answer

Variance

Question 48

Pearson's correlation coefficient measures the strength and direction of the linear relationship between two variables. What is the range of possible values for Pearson's correlation coefficient?

Accepted Answer

-1 to 1

Answer

-0.5 to 0.5

Answer

0 to 1

Answer

-100 to 100

Question 49

Which of the following is a continuous probability distribution used to model the waiting time until a specific event occurs?

Accepted Answer

Exponential distribution

Answer

Normal distribution

Answer

Poisson distribution

Answer

Binomial distribution

Question 50

Which of the following is a key characteristic of a normal distribution?

Accepted Answer

Symmetrical bell-shaped curve

Answer

Positively skewed

Answer

Bimodal

Question 51

In a random sample of 1000 observations from a normally distributed population, the probability of obtaining a sample mean within 1.96 standard deviations of the population mean is approximately:

Accepted Answer

0.95

Answer

0.90

Answer

0.99

Answer

0.80

Question 52

Which of the following statistical tests is used to compare the means of two independent populations?

Accepted Answer

Two-sample t-test

Answer

One-sample proportion test

Answer

Kolmogorov-Smirnov test

Answer

Chi-square test

Question 53

The R-squared value in a regression model represents:

Accepted Answer

The proportion of variance in the dependent variable explained by the independent variable(s)

Answer

The amount of error in the model

Answer

The strength of the relationship between the variables

Question 54

Which of the following is an assumption of the chi-square goodness-of-fit test?

Accepted Answer

The expected frequencies in each category should be at least 5.

Answer

The data must be normally distributed

Answer

The sample size must be small

Question 55

In a hypothesis test, the probability of rejecting the null hypothesis when it is actually true is referred to as:

Accepted Answer

Type I error

Answer

Statistical significance

Answer

Type II error

Question 56

Which of the following sampling methods involves dividing the population into subgroups and randomly selecting a sample from each subgroup?

Accepted Answer

Stratified random sampling

Answer

Convenience sampling

Answer

Cluster sampling

Answer

Quota sampling

Question 57

Which of the following is an advantage of using non-parametric statistical tests?

Accepted Answer

They do not require the data to follow a specific distribution

Answer

They are easier to interpret than parametric tests

Answer

They are more powerful than parametric tests

Question 58

In a multiple regression model with two independent variables, the coefficient of determination (R-squared) represents:

Accepted Answer

The proportion of variance in the dependent variable explained by both independent variables

Answer

The correlation between the independent variables

Answer

The amount of error in the model

Question 59

A sample of 100 students is taken from a population of 1000 students. The sample mean is found to be 75. What is the standard error of the mean to 2 decimal places?

Accepted Answer

2.50

Answer

10

Answer

20

Answer

5

Question 60

Which of the following is a one-sample statistical inference method used to test a hypothesis about a population parameter?

Accepted Answer

Hypothesis testing

Answer

ANOVA

Answer

Regression analysis

Answer

Cluster analysis

Question 61

In a hypothesis test, what is the null hypothesis typically?

Accepted Answer

The hypothesis that there is no significant difference or effect.

Answer

The hypothesis that the two means are equal.

Answer

The hypothesis that there is a significant difference or effect.

Question 62

Which of the following is a non-parametric test, also known as a distribution-free test?

Accepted Answer

Mann-Whitney U test

Answer

ANOVA

Answer

T-test

Answer

Regression analysis

Question 63

Which of the following is a common type of data visualization used to represent categorical data?

Accepted Answer

Bar chart

Answer

Algorithm

Answer

Database

Answer

Programming language

Question 64

What is the fundamental difference between a population and a sample in statistics?

Accepted Answer

A population is the entire group of interest, while a sample is a subset of the population representing the characteristics of the entire population.

Answer

A population is a group of objects, while a sample is a group of events.

Answer

A population is a group of people, while a sample is a group of animals.