Probability and Statistics Quizzes for Engineering Mathematics Masters

Question 1

Which fundamental principle of probability theory dictates that the sum of probabilities for all possible outcomes of an experiment must always equal 1?

Accepted Answer

Normalization Axiom

Answer

Additivity Axiom

Answer

Complement Axiom

Answer

Multiplication Axiom

Question 2

Which fundamental principle of probability theory states that the probability of an event cannot be less than zero?

Accepted Answer

Non-negativity Axiom

Answer

Sample Space Axiom

Answer

Additivity Axiom

Answer

Conditional Probability Axiom

Question 3

Which of the following is a key assumption of the Central Limit Theorem?

Accepted Answer

Observations in the sample must be independent.

Answer

Sample size must be small.

Answer

Underlying population must be normally distributed.

Answer

Sample size must be at least 100.

Question 4

Which of the following is NOT a fundamental principle of probability theory?

Accepted Answer

The probability of an event is unaffected by prior knowledge or information.

Answer

The probability of an event lies between 0 and 1.

Answer

The probability of two events occurring together equals the sum of their individual probabilities.

Answer

The probability of an event is determined by the ratio of favorable outcomes to possible outcomes.

Question 5

Which of the following is a consequence, rather than an axiom, of probability theory?

Accepted Answer

The probability of the union of two events is less than or equal to the sum of their probabilities.

Answer

The probability of an event is non-negative.

Answer

The probability of the sample space is 1.

Answer

The probability of an event is a number between 0 and 1.

Question 6

A random variable X has the following probability mass function: P(X = x) = (1/4)^x for x = 1, 2, 3, .... Calculate the expected value of X.

Accepted Answer

4/3

Answer

1

Answer

1/4

Answer

16/3

Question 7

In a manufacturing process, the probability of producing a defective item is 5%. What is the probability of producing three non-defective items in a row?

Accepted Answer

0.875

Answer

0.008

Answer

0.041

Question 8

In a two-sample hypothesis test, what is the role of the p-value?

Accepted Answer

It indicates the likelihood of obtaining the observed or more extreme results if the null hypothesis is true.

Answer

It measures the probability of making a correct decision in the test.

Answer

It measures the probability of the null hypothesis being true.

Answer

It measures the probability of the alternative hypothesis being true.

Question 9

You have a dataset of 1000 data points and want to use a regression model to predict the value of a new data point. Which approach will likely produce a model that generalizes well to new data?

Accepted Answer

Choose a model with a complexity that balances overfitting and underfitting.

Answer

Use a model with many features to capture all the details of the data.

Answer

Use a model with few features to avoid overfitting.

Question 10

Identify the distribution that is not a valid probability distribution:

Accepted Answer

Cauchy distribution

Answer

Normal distribution

Answer

Uniform distribution

Answer

Exponential distribution

Question 11

Define sampling error and explain how it affects the reliability of statistical conclusions drawn from sample data.

Accepted Answer

Sampling error is the discrepancy between the actual value of a parameter and its estimate derived from a sample due to random sampling.

Answer

Sampling error is insignificant and can be disregarded in statistical inferences.

Answer

Statistical inferences based on sample data are always accurate, regardless of sampling error.

Answer

Sampling error only impacts point estimates, not confidence intervals.

Question 12

What is the probability of obtaining a '6' when rolling a fair dice?

Accepted Answer

1/6

Answer

1/3

Answer

1/4

Answer

1/5

Question 13

What is the distinction between a statistic and a parameter?

Accepted Answer

A statistic is a numerical measure calculated from a sample, while a parameter is a numerical characteristic of a population.

Answer

A statistic is a population value, while a parameter is a sample value.

Answer

A parameter is a population value, while a statistic is a sample value.

Answer

A parameter is a numerical measure calculated from a sample, while a statistic is a numerical characteristic of a population.

Question 14

Which of the options is a measure of central tendency?

Accepted Answer

Mean

Answer

Variance

Answer

Standard deviation

Answer

Coefficient of variation

Question 15

What is the primary purpose of a hypothesis test?

Accepted Answer

To establish whether there is sufficient evidence to reject a hypothesis.

Answer

To confirm the validity of a hypothesis.

Answer

To disprove the validity of a hypothesis.

Answer

To forecast future outcomes.

Question 16

In a regression model, what does the coefficient of determination (R-squared) represent?

Accepted Answer

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

Answer

The slope of the regression line.

Answer

The intercept of the regression line.

Answer

The standard error of the regression.

Question 17

In hypothesis testing, differentiate between Type I and Type II errors. Discuss the factors that impact the likelihood of these errors.

Accepted Answer

Type I error (false positive): Rejecting the null hypothesis when it's true. Type II error (false negative): Failing to reject the null hypothesis when it's false.

Answer

Type I error: Failing to reject the null hypothesis when it's false. Type II error: Rejecting the null hypothesis when it's true.

Answer

Type I and Type II errors are not distinct.

Question 18

Which of the following is NOT a type of continuous probability distribution?

Accepted Answer

Binomial distribution

Answer

Normal distribution

Answer

Exponential distribution

Answer

Log-normal distribution

Question 19

Which of the following is a measure of the strength and direction of the linear relationship between two variables?

Accepted Answer

Correlation coefficient

Answer

Regression coefficient

Answer

Coefficient of determination

Answer

Mean square error

Question 20

A manufacturing process produces items that have a failure rate of 5%. If a random sample of 20 items is selected, what is the probability that exactly 2 items will fail?

Accepted Answer

0.24

Answer

0.0067

Answer

0.067

Answer

0.38

Question 21

What is the probability of rolling a six on a standard die?

Accepted Answer

1/6

Answer

1/5

Answer

1/4

Answer

1/3

Question 22

Which of the following is a measure of central tendency?

Accepted Answer

Mean

Answer

Variance

Answer

Standard deviation

Answer

Range

Question 23

Which of the following is a non-parametric test?

Accepted Answer

Chi-square test

Answer

t-test

Answer

ANOVA

Answer

Regression analysis

Question 24

What is the dependent variable in a linear regression model?

Accepted Answer

The variable that is being predicted

Answer

The variable that is used to predict the dependent variable

Answer

The variable that is held constant

Answer

The variable that is not included in the model

Question 25

Which of the following is NOT an assumption of linear regression?

Accepted Answer

The independent variables are correlated with each other

Answer

The relationship between the dependent and independent variables is linear

Answer

The errors are normally distributed

Answer

The variance of the errors is constant

Question 26

What is the formula for calculating the sample variance?

Accepted Answer

Variance = Sum of squared deviations from the mean / (Number of values - 1)

Answer

Variance = Sum of deviations from the mean / (Number of values - 1)

Answer

Variance = Sum of squares / Number of values

Answer

Variance = Number of values / Sum of squares

Question 27

In a hypothesis test, the null hypothesis proposes that:

Accepted Answer

There is no significant difference between the groups.

Answer

There is a significant difference between the groups.

Answer

The sample mean is significantly greater than the population mean.

Answer

The sample mean is significantly less than the population mean.

Question 28

In the context of quality control, a six-sigma process signifies that:

Accepted Answer

Approximately 99.99966% of products meet specifications.

Answer

Approximately 99.73% of products meet specifications.

Answer

Approximately 95% of products meet specifications.

Answer

Approximately 90% of products meet specifications.

Question 29

Which measure is used to quantify the spread of data:

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Correlation coefficient

Question 30

In a normal distribution, the area under the curve between two z-scores represents:

Accepted Answer

The probability of observing a data point between the two z-scores.

Answer

The mean of the distribution.

Answer

The standard deviation of the distribution.

Answer

The coefficient of variation of the distribution.

Question 31

In a risk assessment, a risk factor is defined as:

Accepted Answer

A factor that increases the probability of an event.

Answer

A factor that decreases the probability of an event.

Answer

A factor that has no influence on the probability of an event.

Answer

A factor that is always present when an event occurs.

Question 32

Given the probability density function f(x) = 1/2 for 0 ≤ x ≤ 2, what is the probability of X taking on a value less than 1?

Accepted Answer

0.25

Answer

0.5

Answer

0.75

Answer

1

Question 33

What does the coefficient of determination (R^2) in linear regression represent?

Accepted Answer

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

Answer

The slope of the regression line

Answer

The y-intercept of the regression line

Answer

The error term of the regression line

Question 34

Which of the following is NOT an example of a discrete probability distribution?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Uniform distribution

Question 35

Which of the following is NOT a measure of spread in a data set?

Accepted Answer

Correlation

Answer

Range

Answer

Standard deviation

Answer

Skewness

Question 36

Which of the following is NOT an assumption of the linear regression model?

Accepted Answer

Errors are normally distributed

Answer

Linear relationship between the dependent and independent variables

Answer

Errors are independent and identically distributed

Answer

Variance of errors is constant

Question 37

In regression analysis, what does the coefficient of determination (R-squared) measure?

Accepted Answer

The percentage of variation in the dependent variable explained by the model

Answer

The strength of the linear relationship between the independent and dependent variables

Answer

The statistical significance of the regression model

Answer

The relative importance of the independent variables compared to the dependent variable

Question 38

Which of the following is the probability mass function of a binomial distribution with n trials and probability of success p?

Accepted Answer

P(X = x) = (n! / (x! * (n-x)!)) * p^x * (1-p)^(n-x)

Answer

P(X = x) = (n! / (x! * (n+x)!)) * p^x * (1-p)^(n+x)

Answer

P(X = x) = (x! / (n! * (n-x)!)) * p^x * (1-p)^(n+x)

Answer

P(X = x) = (x! / (n! * (n+x)!)) * p^x * (1-p)^(n-x)

Question 39

Which of the following is a measure of dispersion for a distribution?

Accepted Answer

Variance

Answer

Mean

Answer

Median

Answer

Mode

Question 40

Which probability distribution is characterized by a bell-shaped curve and is widely used for modeling continuous random variables?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Uniform distribution

Answer

Poisson distribution

Question 41

In hypothesis testing, the critical value refers to:

Accepted Answer

The threshold used to determine whether to reject or fail to reject the null hypothesis

Answer

The probability of obtaining the sample data assuming the null hypothesis is true

Answer

The difference between the sample mean and the hypothesized mean

Question 42

Which assumption is violated in linear regression when there is a non-linear relationship between the independent and dependent variables?

Accepted Answer

Linearity

Answer

Homoscedasticity

Answer

Independence of residuals

Question 43

Which statistical measure quantifies the strength and direction of the linear association between two variables?

Accepted Answer

Correlation coefficient

Answer

Coefficient of determination

Answer

Slope of the regression line

Answer

Standard error of the estimate

Question 44

Which type of sampling bias arises when the sample is not representative of the population due to a systematic error in the sampling method?

Accepted Answer

Selection bias

Answer

Measurement bias

Answer

Response bias

Answer

Non-response bias

Question 45

In a regression model, the coefficient of determination (R²) represents:

Accepted Answer

The proportion of variance in the dependent variable accounted for by the independent variables

Answer

The slope of the regression line

Answer

The intercept of the regression line

Question 46

In a coin tossing experiment, the probability of getting a head is 0.6. What is the probability of NOT getting a head?

Accepted Answer

0.4

Answer

0.8

Answer

0.7

Answer

0.5

Question 47

A linear regression model is fitted to a set of data. The coefficient of determination is 0.8. What is the percentage of the variation in the dependent variable explained by the model?

Accepted Answer

80%

Answer

70%

Answer

60%

Question 48

What is the purpose of a confidence interval?

Accepted Answer

To estimate the true population mean with a given level of confidence.

Answer

To test the hypothesis that the population mean is equal to a specific value.

Answer

To compare the means of two populations.

Question 49

Which of the following is the probability of an event not occurring?

Accepted Answer

P(not event) = 1 - P(event)

Answer

P(event) / P(not event)

Answer

P(event) - P(not event)

Answer

P(not event) / P(event)

Question 50

Which of the following is the formula for the expected value of a random variable X?

Accepted Answer

E(X) = Σ(x * P(x))

Answer

E(X) = Σ(P(x))

Answer

E(X) = Σ(x)

Answer

E(X) = Σ(x^2 * P(x))

Question 51

A sample of 100 students is randomly selected from a population of 1000 students. The sample mean height is 170 cm and the population mean height is 172 cm. What is the standard error of the mean?

Accepted Answer

Standard Error (SE) = 1.22 cm

Answer

SE = 1.73 cm

Answer

SE = 0.88 cm

Question 52

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

Accepted Answer

Two-sample t-test

Answer

Chi-square test

Answer

Analysis of variance (ANOVA)

Answer

Correlation analysis

Question 53

A linear regression analysis is performed between the explanatory variable x and the response variable y. The coefficient of determination (R-squared) is 0.85. What percentage of the variation in y can be explained by the variation in x?

Accepted Answer

85%

Answer

25%

Answer

50%

Answer

15%

Question 54

Which of the following probability distributions is continuous?

Accepted Answer

Normal distribution

Answer

Poisson distribution

Answer

Geometric distribution

Answer

Binomial distribution

Question 55

What is the mean of a Poisson distribution with parameter λ?

Accepted Answer

Mean = λ

Answer

Mean = λ^2

Answer

Mean = λ/2

Answer

Mean = 1/λ

Question 56

A scatter plot of two variables shows a positive correlation. What does this suggest?

Accepted Answer

As one variable increases, the other variable tends to increase.

Answer

There is no relationship between the two variables.

Answer

As one variable increases, the other variable tends to decrease.

Question 57

A construction company is bidding on a project. The company estimates that the probability of winning the bid is 60%, the probability of making a profit if they win is 70%, and the probability of losing money if they lose the bid is 10%. What is the probability of the company making a loss?

Accepted Answer

0.28

Answer

0.30

Answer

0.60

Answer

0.56

Question 58

A regression analysis is performed on data collected from an experiment. The resulting regression equation is: y = -2x + 3. What is the slope of the regression line?

Accepted Answer

-2

Answer

4

Answer

1

Answer

3

Question 59

A correlation coefficient of 0.9 indicates a:

Accepted Answer

Strong positive correlation

Answer

Strong negative correlation

Answer

Weak negative correlation

Answer

Weak positive correlation

Question 60

Which of the following is a measure of dispersion?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Mode

Answer

Median

Question 61

A normal distribution with a mean of 50 and a standard deviation of 8 has a probability of 0.3085 that a randomly selected value will be above what value?

Accepted Answer

57.77

Answer

54.87

Answer

40.87

Answer

45.87

Question 62

What is the probability of rolling a '6' on a standard six-sided die?

Accepted Answer

1/6

Answer

3/8

Answer

1/4

Answer

1/3

Question 63

What is the purpose of using a histogram to represent data?

Accepted Answer

To visualize the distribution and frequency of data points

Answer

To calculate the mean and standard deviation

Answer

To make inferences about the population

Question 64

Which of the following is a type of correlation that measures the strength and direction of the linear relationship between two variables?

Accepted Answer

Pearson's correlation coefficient

Answer

Spearman's rank correlation coefficient

Answer

Coefficient of determination