Advanced Probability and Statistics Learning Quizzes | University Doctor of Philosophy (PhD)

Question 1

Which of the following statements about the Central Limit Theorem is **false**?

Accepted Answer

The CLT requires a small sample size for accurate approximation.

Answer

The CLT applies to sample means of any distribution.

Answer

The CLT states that the distribution of sample means approaches a normal distribution as the sample size increases.

Answer

The CLT can be used to test hypotheses about the population mean.

Question 2

In a chi-squared test for independence, what is the purpose of the chi-squared statistic?

Accepted Answer

Determine if the observed and expected frequencies of outcomes differ significantly

Answer

Calculate the probability of obtaining the observed data if the null hypothesis is true

Answer

Test the assumption that the variables are normally distributed

Answer

Estimate the variance of the sampling distribution of the sample proportion

Question 3

In a binomial distribution with n = 10 and p = 0.5, what's the probability of X being greater than 6?

Accepted Answer

0.0547

Answer

0.2059

Answer

0.9453

Answer

0.4951

Question 4

Given a fair six-sided die, where X represents the number on the top face, determine the expected value of X.

Accepted Answer

3.5

Answer

1

Answer

2

Answer

5

Question 5

In probability theory, which formula calculates the probability of the union of events A and B?

Accepted Answer

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Answer

P(A ∪ B) = P(A) - P(B)

Answer

P(A ∪ B) = P(A) * P(B)

Answer

P(A ∪ B) = P(A) / P(B)

Question 6

Which of the following is not a core component of Bayesian inference?

Accepted Answer

Sampling distribution

Answer

Prior probability distribution

Answer

Likelihood function

Answer

Posterior probability distribution

Question 7

In the context of sampling distributions, what is the term used to describe the standard deviation of the sampling distribution of the mean?

Accepted Answer

Standard error of the mean

Answer

Margin of error

Answer

Standard deviation of the mean

Answer

Sampling error

Question 8

For independent events A and B, which formula correctly calculates their joint probability, P(A ∩ B)?

Accepted Answer

P(A ∩ B) = P(A) * P(B)

Answer

P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

Answer

P(A ∩ B) = P(A) - P(B)

Answer

P(A ∩ B) = P(A) / P(B)

Question 9

In ANOVA, what is the main purpose of using multiple comparison procedures?

Accepted Answer

Adjust p-values for individual hypothesis tests to control for multiple comparisons and reduce Type I errors.

Answer

Increase statistical power by enlarging sample sizes for each hypothesis, reducing Type II errors.

Answer

Simplify ANOVA results by combining p-values from individual hypothesis tests into a single overall p-value.

Question 10

For a continuous random variable X with probability density function f(x), which of the following is true about its cumulative distribution function F(x)?

Accepted Answer

F(x) is a non-decreasing function.

Accepted Answer

F(x) is the integral of f(x).

Answer

F(x) is always equal to 1.

Question 11

Which of the following is NOT a characteristic of a normal distribution?

Accepted Answer

The mean, median, and mode are always equal.

Answer

Its graph is bell-shaped and symmetrical.

Answer

Probabilities for a specific range of values can be calculated using the standard normal distribution.

Answer

The total area under the normal curve is 1, representing the probability of all possible outcomes.

Question 12

In post-hoc tests following an ANOVA, which adjustment is typically used to control the overall probability of a Type I error?

Accepted Answer

Bonferroni correction

Answer

Tukey's HSD test

Answer

Dunn's test

Answer

Scheffe's test

Question 13

For a continuous random variable, which measure represents the average value it is likely to take?

Accepted Answer

Expected value (mean)

Answer

Mode

Answer

Variance

Question 14

Which of the following is **not** an essential assumption for the validity of a linear regression model?

Accepted Answer

Absence of multicollinearity among independent variables

Answer

Linear relationship between dependent and independent variables

Answer

Constant variance of residuals

Answer

Uncorrelated errors (residuals)

Question 15

In a study involving categorical data, how do we determine if there is an association between two categorical variables using the chi-square test of independence?

Accepted Answer

Calculate the expected frequencies for each cell in the contingency table and compare them to the observed frequencies.

Answer

Calculate the correlation coefficient between the two variables.

Answer

Use a t-test to compare the means of the two variables.

Answer

Use an analysis of variance (ANOVA) to compare the variances of the two variables.

Question 16

In hypothesis testing, what is the probability of committing a Type I error?

Accepted Answer

The probability of rejecting a true null hypothesis.

Answer

The probability of failing to reject a false null hypothesis.

Answer

The probability of accepting a false null hypothesis.

Answer

The probability of rejecting an alternative hypothesis.

Question 17

Which of the following is a non-parametric test for comparing two independent samples?

Accepted Answer

Mann-Whitney U test

Answer

t-test

Answer

ANOVA

Answer

Pearson chi-squared test

Question 18

What does the central limit theorem state?

Accepted Answer

For large sample sizes, the distribution of sample means approaches normality.

Answer

For large sample sizes, the sample variance approaches the population variance.

Answer

For all sample sizes, the distribution of individual data points is approximately normal.

Answer

For large sample sizes, the probability of an event approaches its expected value.

Question 19

The probability density function of a standard normal distribution is given by which of the following?

Accepted Answer

f(x) = (1/sqrt(2π)) * e^(-x^2/2)

Answer

f(x) = (1/√(2π)) * e^(x^2/2)

Answer

f(x) = (1/√(π)) * e^(-x^2/2)

Answer

f(x) = (1/√(2π)) * e^(x^2)

Question 20

Which sampling method ensures that every member of the population has an equal chance of being selected?

Accepted Answer

Simple random sampling

Answer

Stratified sampling

Answer

Cluster sampling

Answer

Convenience sampling

Question 21

Which of the following measures the spread or dispersion of a distribution?

Accepted Answer

Variance

Answer

Mean

Answer

Median

Answer

Mode

Question 22

Which type of random variable can take on a continuous range of values?

Accepted Answer

Continuous

Answer

Discrete

Answer

Categorical

Answer

Ordinal

Question 23

What is the purpose of a confidence interval in statistical inference?

Accepted Answer

To estimate the population mean or proportion

Answer

To test a hypothesis

Answer

To describe the distribution of a sample

Answer

To determine a sample size

Question 24

How does increasing the sample size affect the width of a confidence interval?

Accepted Answer

Decreases

Answer

Increases

Answer

No effect

Answer

Depends on the standard deviation

Question 25

Which of the following is a property of the binomial distribution?

Accepted Answer

Mean = np and Variance = np(1-p)

Answer

Discrete and symmetric

Answer

Mean and variance are both equal to np

Answer

Symmetric and has a variance of np(1-p)

Question 26

A random variable X has the following probability mass function: P(X=x) = (1/2)^x for x = 0, 1, 2, .... What type of distribution does X follow?

Accepted Answer

Geometric distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Negative binomial distribution

Question 27

What is the critical value for a two-sided z-test with a level of significance of 0.05?

Accepted Answer

1.96

Answer

1.645

Answer

2.326

Answer

2.576

Question 28

Which of the following is a measure of dispersion?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Mode

Question 29

A scatterplot shows a negative correlation between two variables. What does this indicate?

Accepted Answer

As one variable increases, the other variable decreases.

Answer

As one variable increases, the other variable also increases.

Answer

There is no relationship between the two variables.

Answer

The relationship between the two variables is non-linear.

Question 30

Which of the following is a parametric test for comparing the means of two independent samples?

Accepted Answer

t-test

Answer

ANOVA

Answer

Mann-Whitney U test

Answer

Kruskal-Wallis test

Question 31

Define effect size and explain its significance in determining the appropriate sample size for a statistical study.

Accepted Answer

Effect size quantifies the magnitude of an effect and is crucial for calculating the necessary sample size for a study.

Answer

Effect size is exclusively applicable to qualitative research methods.

Answer

Sample size determination can be made without considering effect size.

Answer

Effect size is directly used to calculate statistical power.

Question 32

What is a key benefit of using Monte Carlo simulation in complex statistical scenarios?

Accepted Answer

It can be applied to a variety of probability distributions.

Answer

It guarantees an exact solution to any statistical problem.

Answer

It requires minimal computational resources.

Question 33

How does the central limit theorem enable us to draw inferences about population parameters using sample statistics?

Accepted Answer

It allows us to make inferences about population parameters based on sample statistics.

Answer

It guarantees that sample means will always follow a normal distribution.

Answer

It reduces the variation in sample estimates.

Answer

It eliminates the necessity for random sampling.

Question 34

Which technique is specifically designed to analyze time-series data, accounting for its autocorrelation and seasonality?

Accepted Answer

ARIMA modeling

Answer

Regression analysis

Answer

Analysis of variance (ANOVA)

Answer

Bayesian statistics

Question 35

What is the probability of obtaining a value less than or equal to 2 in a standard normal distribution?

Accepted Answer

0.9772

Answer

0.95

Answer

0.05

Answer

0.0228

Question 36

Which of the following sampling methods is least likely to produce a representative sample of a population?

Accepted Answer

Convenience sampling

Answer

Cluster sampling

Answer

Quota sampling

Answer

Simple random sampling

Question 37

Which of the following tests is used to determine the significance of the difference between two sample variances?

Accepted Answer

F-test

Answer

t-test

Answer

Chi-square test

Question 38

What is the type II error rate?

Accepted Answer

The probability of accepting the null hypothesis when it is false.

Answer

The probability of rejecting the null hypothesis when it is true.

Answer

The probability of rejecting the null hypothesis when it is false.

Question 39

What is the purpose of a logistic regression model?

Accepted Answer

To predict the probability of a binary outcome.

Answer

To predict the value of a continuous outcome.

Answer

To describe the relationship between two or more variables.

Question 40

Which of the following statements is NOT a characteristic of a probability distribution?

Accepted Answer

The mean is always positive.

Answer

The distribution can be represented graphically.

Answer

The probability of any outcome is between 0 and 1.

Answer

The sum of the probabilities of all outcomes is 1.

Question 41

Which of the following is a measure of the spread or dispersion of a dataset?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Kurtosis

Answer

Skewness

Question 42

To test for a significant difference between the means of two independent groups, which hypothesis test is appropriate?

Accepted Answer

Two-sample t-test

Answer

One-sample t-test

Answer

Chi-square test

Answer

ANOVA

Question 43

What is a fundamental limitation of statistical inference?

Accepted Answer

It provides estimates and probabilities, not absolute certainty.

Answer

It is only applicable to quantitative data.

Answer

It is not always easy to interpret.

Answer

It requires a large sample size.

Question 44

A researcher observes a strong positive correlation between age and income in a sample of 100 individuals. Can they definitively conclude that age causes income?

Accepted Answer

No, correlation does not imply causation.

Answer

Maybe, further investigation is needed.

Answer

Yes, a strong correlation suggests causation.