Statistics for Year 13: Master Regression, Probability, and Hypothesis Testing

Question 1

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

Accepted Answer

Correlation

Answer

Mean

Answer

Median

Answer

Mode

Question 2

In queueing theory, which probability distribution is used to model the time between arrivals?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Normal distribution

Answer

Poisson distribution

Question 3

In hypothesis testing, a Type I error occurs when:

Accepted Answer

We reject a true null hypothesis

Answer

We fail to reject a false null hypothesis

Answer

We accept a true null hypothesis

Answer

We accept a false null hypothesis

Question 4

In a two-sample t-test, what is the null hypothesis?

Accepted Answer

The means of the two samples are equal

Answer

The means of the two samples are different

Answer

The first sample mean is greater than the second sample mean

Answer

The second sample mean is greater than the first sample mean

Question 5

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

Accepted Answer

Chi-squared test

Answer

Pearson's correlation coefficient

Answer

Linear regression

Answer

Analysis of variance

Question 6

What is the primary purpose of a scatterplot?

Accepted Answer

To visualize the relationship between two variables

Answer

To compare the means of two groups

Answer

To test the statistical significance of a correlation

Answer

To predict the value of one variable based on another

Question 7

In linear regression, the coefficient of determination (R-squared) indicates:

Accepted Answer

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

Answer

The strength of the relationship between the independent and dependent variables

Answer

The p-value for the F-test

Answer

The slope of the regression line

Question 8

Which probability distribution is used to model continuous random variables bounded between 0 and 1?

Accepted Answer

Uniform distribution

Answer

Binomial distribution

Answer

Normal distribution

Answer

Poisson distribution

Question 9

A key assumption of the central limit theorem is that:

Accepted Answer

The sample size is sufficiently large (typically n > 30)

Answer

The population distribution follows a normal distribution

Answer

The sampling method is random

Answer

The variance of the population is known

Question 10

When a researcher anticipates a non-linear relationship, which regression model is most appropriate?

Accepted Answer

Polynomial regression

Answer

Linear regression

Answer

Exponential regression

Answer

Logistic regression

Question 11

The probability of obtaining a sample mean within a certain range of the actual population mean is termed:

Accepted Answer

Confidence interval

Answer

Sampling error

Answer

Standard error

Answer

Margin of error

Question 12

A model utilizing the independent variable to forecast the mean of the dependent variable is known as a:

Accepted Answer

Linear regression model

Answer

Exponential regression model

Answer

Logistic regression model

Answer

Multiple regression model

Question 13

A probability distribution describing the distribution of sample means is known as a:

Accepted Answer

Sampling distribution

Answer

Chi-square distribution

Answer

Normal distribution

Answer

Binomial distribution

Question 14

Which non-parametric test is used specifically to compare medians?

Accepted Answer

Mann-Whitney U test

Answer

Wilcoxon signed-rank test

Answer

Kruskal-Wallis test

Answer

Friedman test

Question 15

The expected value of a random variable X can be expressed as:

Accepted Answer

The total of all probable values of X multiplied by their respective probabilities

Answer

The mathematical mean of the probability distribution of X

Answer

The probability of X attaining its highest possible value

Answer

The probability of X attaining its lowest possible value

Question 16

A Pearson correlation coefficient of 0.8 indicates:

Accepted Answer

A strong positive linear relationship between the two variables.

Answer

A strong negative linear relationship between the two variables.

Answer

A weak positive linear relationship between the two variables.

Answer

No linear relationship between the two variables.

Question 17

In hypothesis testing, the null hypothesis assumes that:

Accepted Answer

The difference between groups is not significant.

Answer

The alternative hypothesis is true.

Answer

The data follows a specific distribution.

Answer

The difference between groups is significant.

Question 18

When flipping a fair coin twice, the probability of getting a head both times is:

Accepted Answer

1/4

Answer

1/2

Answer

3/4

Answer

1

Question 19

The formula for the slope of a linear regression line is:

Accepted Answer

(Σ(x - x̄)(y - ȳ)) / Σ(x - x̄)²

Answer

(Σ(x - ȳ)(y - x̄)) / Σ(x - ȳ)²

Answer

(Σ(y - x̄)(x - ȳ)) / Σ(y - x̄)²

Answer

(Σ(y - ȳ)(x - x̄)) / Σ(y - ȳ)²

Question 20

The difference between Type I and Type II errors in hypothesis testing is:

Accepted Answer

Type I error: rejecting the null hypothesis when it is true; Type II error: accepting the null hypothesis when it is false.

Answer

Type I error: accepting the null hypothesis when it is false; Type II error: rejecting the null hypothesis when it is true.

Answer

Both Type I and Type II errors involve rejecting the null hypothesis.

Answer

Both Type I and Type II errors involve accepting the null hypothesis.

Question 21

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

Accepted Answer

Skewness

Answer

Linearity

Answer

Homoscedasticity

Answer

Independence

Question 22

In hypothesis testing, which factor does NOT affect the level of statistical significance?

Accepted Answer

Measurement error

Answer

Sample size

Answer

Significance level

Question 23

Which sampling technique guarantees that each individual in a population has an equal opportunity to be chosen, reducing the likelihood of sampling bias?

Accepted Answer

Random sampling

Answer

Voluntary response

Answer

Convenience sampling

Answer

Quota sampling

Question 24

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

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Mode

Question 25

In a normal distribution, the area under the curve between two z-scores can be calculated using the:

Accepted Answer

Standard normal distribution table

Answer

T-distribution table

Answer

Chi-square distribution table

Answer

F-distribution table

Question 26

Which of the following statistical models is appropriate for predicting a continuous dependent variable based on one or more independent variables?

Accepted Answer

Linear regression

Answer

Logistic regression

Answer

Poisson regression

Answer

ANOVA

Question 27

The p-value in a hypothesis test represents the:

Accepted Answer

Probability of obtaining the observed results assuming the null hypothesis is true

Answer

Probability of rejecting the null hypothesis when it is true

Answer

Probability of accepting the null hypothesis when it is false

Answer

Probability of making a Type I error

Question 28

The coefficient of determination (R-squared) in linear regression represents the:

Accepted Answer

Proportion of variance in the dependent variable explained by the independent variables

Answer

Slope of the regression line

Answer

Y-intercept of the regression line

Answer

Correlation coefficient

Question 29

Write the formula for calculating the mean of a sample.

Accepted Answer

Mean = Σx / n

Answer

Mean = n / Σx

Answer

Mean = Σ(x - x̄) / n

Answer

Mean = Σx^2 / n

Question 30

A random sample of 100 students reveals a correlation coefficient of -0.8 between height and weight. What can you conclude?

Accepted Answer

Height and weight have a strong inverse linear relationship.

Answer

Height and weight have a strong positive linear relationship.

Answer

Height and weight have a weak inverse linear relationship.

Answer

Height and weight have no linear relationship.

Question 31

Which test is used to determine if there is a significant difference between two means?

Accepted Answer

t-test

Answer

Chi-square test

Answer

ANOVA

Answer

F-test

Question 32

What is the purpose of constructing a confidence interval?

Accepted Answer

To estimate a true population parameter with a certain level of confidence.

Answer

To test the significance of a statistical hypothesis.

Answer

To predict future values of a variable.

Answer

To describe the distribution of a random variable.

Question 33

Which of the following is NOT a step in the statistical modeling process?

Accepted Answer

Experimentation

Answer

Problem formulation

Answer

Data collection

Answer

Model evaluation

Question 34

Define the difference between a population parameter and a sample statistic.

Accepted Answer

A parameter is a characteristic of an entire population, while a statistic is a characteristic of a sample drawn from that population.

Answer

A statistic is a characteristic of an entire population, while a parameter is a characteristic of a sample drawn from that population.

Answer

Parameters are known values, while statistics are estimated values.

Answer

Statistics are known values, while parameters are estimated values.

Question 35

In a scatter plot, the line of best fit has a positive slope. What can be concluded?

Accepted Answer

There is a positive correlation between the two variables.

Answer

There is a negative correlation between the two variables.

Answer

The two variables are not correlated.

Answer

The line of best fit is a poor representation of the data.

Question 36

A survey aims to determine the average height of students in an institute. A sample of 100 students is measured. Which value represents the population parameter?

Accepted Answer

The average height of all students in the institute.

Answer

The average height of the sample.

Answer

The height of the tallest student in the institute.

Answer

The height of the shortest student in the institute.

Question 37

Which measure quantifies the spread of data?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Range

Question 38

In a regression model predicting sales based on advertising expenditure, which variable is dependent?

Accepted Answer

Sales

Answer

Advertising expenditure

Answer

Time

Answer

Product price

Question 39

In a chi-square test for association between categorical variables, what is the null hypothesis?

Accepted Answer

There is no significant association between the variables.

Answer

There is a significant association between the variables.

Answer

The variables are independent.

Answer

The variables are dependent.

Question 40

Which of the following is NOT a measure of central tendency, which provides a numerical value to summarize the middle or center of a dataset?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Mode

Question 41

When flipping a fair coin, what is the probability of obtaining a head, assuming equal chances of either outcome?

Accepted Answer

0.5

Answer

0.25

Answer

0.75

Answer

1

Question 42

Which statistical model is used to predict a continuous outcome variable, such as predicting house prices based on various factors?

Accepted Answer

Linear regression

Answer

Logistic regression

Answer

Poisson regression

Answer

Negative binomial regression

Question 43

In linear regression, what does the slope of the regression line represent, providing insight into the relationship between the predictor and outcome variables?

Accepted Answer

The change in the outcome variable for a one-unit change in the predictor variable

Answer

The intercept of the regression line

Answer

The correlation coefficient

Answer

The standard error of the slope

Question 44

Describe the key difference between a confidence interval and a prediction interval.

Accepted Answer

Confidence intervals estimate population parameters, while prediction intervals estimate future observations.

Answer

Confidence intervals are wider than prediction intervals.

Answer

Prediction intervals are wider than confidence intervals.

Answer

Confidence intervals and prediction intervals are identical.

Question 45

Which formula accurately calculates the Pearson correlation coefficient?

Accepted Answer

r = (Σxy - ΣxΣy) / √[(Σx² - (Σx)²)Σ(y² - (Σy)²)]

Answer

r = Σ(x - x̄)(y - ȳ) / √[Σ(x - x̄)²Σ(y - ȳ)²]

Answer

r = (Σ(x - x̄)(y - ȳ)) / √[(Σ(x - x̄)²) - Σ(y - ȳ)²]

Answer

r = (ΣxΣy - Σxy) / √[(Σx² + Σy²) - (ΣxΣy)]

Question 46

Distinguish between a population and a sample.

Accepted Answer

A population represents the entire group, while a sample is a smaller subset.

Answer

A sample represents the entire group, while a population is a smaller subset.

Answer

A population is hypothetical, while a sample is real.

Answer

A sample is hypothetical, while a population is real.

Question 47

What is the fundamental purpose of hypothesis testing?

Accepted Answer

To determine if evidence supports rejecting a null hypothesis.

Answer

To establish the validity of a null hypothesis.

Answer

To generate new data.

Answer

To predict future events.

Question 48

Distinguish between biased and unbiased estimators.

Accepted Answer

Biased estimators consistently deviate from the true parameter value, while unbiased estimators do not.

Answer

Unbiased estimators consistently deviate from the true parameter value, while biased estimators do not.

Answer

Biased estimators are always more accurate than unbiased estimators.

Answer

Unbiased estimators are always more accurate than biased estimators.

Question 49

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

Accepted Answer

1/6

Answer

1/2

Answer

1/3

Answer

1/4

Question 50

In regression analysis, which step primarily evaluates the practical significance of the relationship between variables?

Accepted Answer

Effect Size Calculation

Answer

Hypothesis Testing

Answer

Model Building

Answer

Residual Analysis

Question 51

Which non-parametric test is used to determine the difference in medians between two independent groups?

Accepted Answer

Mann-Whitney U test

Answer

One-sample t-test

Answer

Two-sample t-test

Answer

Wilcoxon signed-rank test

Question 52

Which characteristic does NOT apply to a normal distribution?

Accepted Answer

Skewness

Answer

Bell-shaped

Answer

Symmetrical

Answer

Unimodal

Question 53

When designing a statistical study, which of the following ethical considerations should researchers prioritize?

Accepted Answer

Obtaining informed consent from participants and ensuring transparency in data collection methods.

Answer

Publicly disseminating research findings to inform the general public about statistical advancements.

Answer

Employing the most sophisticated statistical techniques available, regardless of their ethical implications.

Answer

Manipulating data to achieve desired outcomes without facing ethical consequences.

Question 54

What is a key benefit of using statistical software?

Accepted Answer

Statistical software automates complex calculations, freeing up statisticians to focus on interpreting and drawing conclusions from data.

Answer

All statistical software is equally effective for all types of statistical analysis.

Answer

Open-source statistical software is always better than proprietary statistical software.

Question 55

What is the primary purpose of a correlation coefficient?

Accepted Answer

To measure the strength and direction of the relationship between two variables.

Answer

To test whether there is a significant difference between two groups.

Answer

To predict the value of one variable from the value of another variable.

Question 56

Which of the following is an example of a discrete probability distribution?

Accepted Answer

Binomial distribution

Answer

Exponential distribution

Answer

Uniform distribution

Answer

Normal distribution

Question 57

What is the expected value of a random variable?

Accepted Answer

The mean of the probability distribution.

Answer

The median of the probability distribution.

Answer

The standard deviation of the probability distribution.

Answer

The mode of the probability distribution.

Question 58

What is the main purpose of a hypothesis test?

Accepted Answer

To determine whether there is sufficient evidence to reject the null hypothesis.

Answer

To prove that the alternative hypothesis is true.

Answer

To estimate the population mean.

Question 59

What is the primary purpose of a statistical model?

Accepted Answer

To represent a real-world phenomenon and make predictions.

Answer

To test hypotheses

Answer

To describe a population

Question 60

Which measure of central tendency describes the most commonly occurring value in a data set?

Accepted Answer

Mode

Answer

Mean

Answer

Range

Answer

Median

Question 61

When rolling a standard six-sided die, what is the probability of obtaining an even number?

Accepted Answer

1/2

Answer

1/5

Answer

1/4

Answer

1/3

Question 62

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

Accepted Answer

The sample size must be large enough (typically n>30).

Answer

The data must be perfectly normally distributed.

Answer

The sampling method must be systematic.

Answer

The population standard deviation must be known.

Question 63

In a hypothesis test, the null hypothesis is rejected if the p-value is:

Accepted Answer

Less than the significance level (α).

Answer

Greater than the significance level (α).

Answer

Always rejected, regardless of the p-value.

Answer

Equal to the significance level (α).

Question 64

Which of the following is an example of a parametric statistical model?

Accepted Answer

Linear regression model

Answer

Scatter plot

Answer

Box plot