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What is the mean of the data set {2, 4, 6, 8, 10}?
6
4
7
5
The mean is calculated by summing all values and dividing by the number of observations: (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6. This measure represents the arithmetic average of the data set. It is sensitive to every value, including outliers. More on the arithmetic mean.
What is the median of the data set {3, 7, 8, 10, 12}?
9
7
10
8
The median is the middle value when the data set is ordered. Here, the ordered set is {3, 7, 8, 10, 12}, so the median is 8. It is less affected by extreme values than the mean. More on the median.
Which measure of central tendency is most influenced by outliers?
Mode
Range
Mean
Median
The mean uses all data values in its calculation, so extreme values (outliers) can significantly shift it. In contrast, the median and mode are more robust to outliers. Learn more.
What is the mode of the data set {1, 2, 2, 3, 4, 4, 4, 5}?
3
5
2
4
The mode is the value that appears most frequently in a data set. Here, 4 appears three times, more than any other value. More on mode.
What is the range of the data set {5, 10, 15, 20}?
10
20
15
5
The range is the difference between the maximum and minimum values: 20 - 5 = 15. It gives a simple measure of dispersion. More on range.
Which of the following best describes variance?
Difference between max and min
Square root of standard deviation
Average squared deviation from the mean
Median of the data set
Variance measures dispersion by averaging the squared deviations of each observation from the mean. It is fundamental for other statistics like standard deviation. More on variance.
A variable measured on a nominal scale is:
Ordered categories
Labels without order
Ratios with a true zero
Numeric continuous values
Nominal scales categorize data without any inherent order. Examples include gender, eye color, or blood type. More on measurement scales.
Which of the following is a discrete variable?
Weight in kilograms
Height in centimeters
Temperature
Number of students in a class
A discrete variable takes integer values that can be counted, like the number of students. Continuous variables can take any value within a range. More on variable types.
What does a probability value of 0.75 indicate?
Event is unlikely
Event is likely
Event is certain
Event is impossible
A probability of 0.75 means there is a 75% chance the event will occur, indicating it is likely. Probabilities range between 0 and 1. Learn more.
If P(A) = 0.3, what is P(not A)?
0.3
0.7
1.3
0.5
The probability of the complement of A is 1 ? P(A): 1 ? 0.3 = 0.7. Complements cover all outcomes not in event A. More on complements.
Two events are mutually exclusive if:
One causes the other
They always occur together
Their probabilities sum to less than 1
They cannot occur together
Mutually exclusive events cannot happen at the same time; their intersection is empty. For such events, P(A and B)=0. More on mutually exclusive events.
What is the expected value of a fair die roll?
2.5
4.5
3.0
3.5
The expected value is the average of all possible outcomes: (1+2+3+4+5+6)/6 = 21/6 = 3.5. It represents the long-run mean. More on expected value.
Which of the following describes the standard deviation?
Square root of variance
Variance squared
Average deviation from median
Range divided by 2
Standard deviation is the square root of the variance, bringing the dispersion measure back to the original units. It quantifies average distance from the mean. More on standard deviation.
Which level of measurement has a true zero point and equal intervals?
Nominal
Ordinal
Ratio
Interval
Ratio scales have equal intervals and a meaningful zero, allowing statements about how many times greater one value is than another. Examples include weight and height. More on levels.
Which of the following best defines a population parameter?
A value that describes an entire population
An outlier in the data
A prediction for future data
A statistic calculated from sample data
A parameter is a numerical characteristic of a population, like the population mean or variance. Statistics estimate parameters from sample data. More on parameters.
What is the z-score of X=60 in a normal distribution with mean 50 and standard deviation 5?
2.0
1.5
2.5
1.0
The z-score is calculated as (X??)/? = (60?50)/5 = 10/5 = 2. However, none of those match except if rounding differently. Wait, 10/5=2. So correct answer should be 2. Score for 2.0 should be 1. Let's correct answers.
According to the empirical rule, approximately what percentage of data lies within one standard deviation of the mean in a normal distribution?
95%
99.7%
50%
68%
Empirical rule states about 68% of data in a normal distribution falls within ±1 standard deviation of the mean. ±2 sd covers 95%, ±3 sd covers 99.7%. More on the empirical rule.
Which theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases?
Chebyshev's Inequality
Central Limit Theorem
Law of Large Numbers
Bayes' Theorem
The Central Limit Theorem says that, regardless of the population distribution, the distribution of sample means will approximate normality as n grows large. This underpins many inferential methods. More on CLT.
When the population standard deviation is known, which distribution is used to construct a confidence interval for the mean?
Chi-square distribution
t-distribution
F-distribution
Standard normal distribution
If the population standard deviation is known, the z (standard normal) distribution is used for the confidence interval. If unknown, the t-distribution applies. More on z-intervals.
Which distribution is appropriate for modeling the count of rare events in a fixed interval?
Normal distribution
Poisson distribution
Uniform distribution
Binomial distribution
The Poisson distribution models the number of rare events occurring in a fixed interval when events occur independently and the average rate is constant. More on Poisson.
What is the standard error of the mean if ?=10 and n=25?
0.4
2
10
5
Standard error is ?/?n = 10/?25 = 10/5 = 2. It measures the variability of sample means around the population mean. More on standard error.
In hypothesis testing, the p-value represents:
The probability the alternative hypothesis is true
The probability of observing data as extreme as the sample, assuming the null is true
The significance level
The probability the null hypothesis is true
The p-value is the probability of obtaining test results at least as extreme as those observed, assuming the null hypothesis is true. It does not give the probability that the null is true. More on p-values.
A Type I error occurs when:
Accept the alternative when true
Fail to reject a false null hypothesis
Reject a false null hypothesis
Reject a true null hypothesis
A Type I error is the incorrect rejection of a true null hypothesis, also called a false positive. Its probability is the significance level ?. More on errors.
Which test would you use to compare the means of two independent samples with unknown but equal variances?
Chi-square test
ANOVA
Paired t-test
Two-sample t-test
A two-sample t-test (pooled) compares means of two independent groups when variances are assumed equal but unknown. If paired, use a paired t-test. More on t-tests.
Which rule allows a binomial distribution to be approximated by a normal distribution?
n is large and p is near 0.5
n < 30
p is near 1
n is small and p is near 0
The normal approximation to the binomial is reasonable when n is large and p is not too close to 0 or 1 (commonly np and n(1?p) both ?10). More on approximation.
What is the primary purpose of a chi-square goodness-of-fit test?
Assess how well observed frequencies match expected frequencies
Estimate regression coefficients
Test association between two continuous variables
Compare sample mean to population mean
A chi-square goodness-of-fit test assesses whether observed categorical data match an expected distribution. It compares observed and expected counts. More on chi-square tests.
Which test is appropriate for comparing means across three or more independent groups?
Correlation
t-test
Chi-square test
ANOVA
ANOVA (Analysis of Variance) compares means across three or more groups by analyzing variances within and between groups. It uses the F-distribution. More on ANOVA.
Which assumption must hold for one-way ANOVA?
Homogeneity of variances
Binary outcome
Multicollinearity
Zero skewness
One-way ANOVA assumes homogeneity of variances (equal variances) across groups, independence of observations, and normally distributed residuals. Violation can affect Type I error. More on ANOVA assumptions.
What does the Pearson correlation coefficient measure?
Association between ranks
Causal relationship between variables
Difference in group means
Strength and direction of a linear relationship
The Pearson correlation coefficient (r) quantifies the strength and direction of a linear relationship between two continuous variables, ranging from ?1 to +1. More on Pearson's r.
In simple linear regression, what does R-squared represent?
Intercept value
Proportion of variance explained by the model
Variance of residuals
Slope of the line
R-squared indicates the proportion of variance in the dependent variable that is predictable from the independent variable(s). It ranges from 0 to 1. More on R-squared.
What does multicollinearity refer to in multiple regression?
Non-normal residuals
Heteroskedasticity
High correlation among independent variables
Outliers in the data
Multicollinearity occurs when independent variables in a regression are highly correlated, making it difficult to estimate individual coefficients reliably. More on multicollinearity.
Which test checks for heteroskedasticity in regression residuals?
Durbin-Watson test
Breusch-Pagan test
Levene's test
Shapiro-Wilk test
The Breusch-Pagan test assesses whether the variance of residuals from a regression is constant (homoskedastic). Rejecting indicates heteroskedasticity. More on Breusch-Pagan.
What principle states that sample means converge to the population mean as sample size increases?
Chebyshev's Inequality
Bayes' Theorem
Central Limit Theorem
Law of Large Numbers
The Law of Large Numbers says that as sample size increases, the sample mean approaches the population mean, ensuring stability in long-run averages. More on LLN.
Maximum likelihood estimation is used to:
Minimize the sum of squared errors
Compute Bayesian posterior distributions
Calculate confidence intervals
Find parameter values that maximize the probability of observed data
MLE selects parameter values that maximize the likelihood of the observed sample under the assumed statistical model. It is foundational in statistical inference. More on MLE.
Which of the following is a nonparametric test for comparing two independent samples?
Wilcoxon rank-sum test
Paired t-test
Z-test
Independent t-test
The Wilcoxon rank-sum test (Mann-Whitney U) compares two independent samples without assuming normality. It uses ranks instead of raw data. More on Wilcoxon.
Which test extends the Wilcoxon rank-sum to more than two groups?
Friedman test
Kruskal-Wallis test
ANOVA
Chi-square test
The Kruskal-Wallis test is a nonparametric alternative to one-way ANOVA for comparing more than two groups on an ordinal or continuous outcome. More on Kruskal-Wallis.
What does the Cramér - Rao lower bound provide in estimation theory?
Bayesian posterior variance
Minimum variance bound for unbiased estimators
Maximum likelihood estimate
Maximum possible variance of an estimator
The Cramér - Rao lower bound gives the lowest variance that any unbiased estimator of a parameter can achieve, setting a benchmark for efficiency. More on Cramér - Rao.
In a generalized linear model, the link function:
Transforms predictors to response scale
Ensures normality of residuals
Transforms the response to a linear predictor scale
Is only used in linear regression
The link function connects the expected value of the response variable to the linear predictor, allowing modeling of non-normal outcomes. More on GLMs.
Gibbs sampling is a type of:
Optimization algorithm
Markov chain Monte Carlo method
Direct integration
Nonparametric test
Gibbs sampling is an MCMC technique that generates samples from a multivariate distribution by sampling each variable conditional on the others. More on Gibbs sampling.
The Metropolis-Hastings algorithm is used for:
Estimating p-values
Calculating confidence intervals
Solving linear equations
Sampling from complex probability distributions
Metropolis-Hastings is an MCMC method to obtain a sequence of dependent samples from a target distribution when direct sampling is difficult. More on Metropolis-Hastings.
Empirical Bayes methods:
Ignore prior information
Require fully known priors
Are only for nonparametric inference
Use data to estimate prior distributions
Empirical Bayes estimates the prior distribution parameters from the data itself, blending Bayesian and frequentist ideas for more robust inference. More on Empirical Bayes.
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Study Outcomes
Understand Foundational Probability Distributions -
Identify and describe the key characteristics of normal, binomial, and Poisson distributions to solve relevant statistics problems confidently.
Apply Hypothesis Testing Procedures -
Execute each step of hypothesis testing, including formulating null and alternative hypotheses and calculating p-values for decision making.
Calculate Descriptive Statistics -
Compute measures of central tendency and variability - such as mean, median, variance, and standard deviation - to summarize data sets effectively.
Interpret Statistical Results -
Analyze confidence intervals and test outcomes to draw meaningful conclusions and assess the strength of evidence.
Evaluate Test Assumptions -
Assess underlying conditions like normality and independence to ensure the validity of statistical tests in multiple-choice scenarios.
Develop MCQ Test-Taking Strategies -
Implement proven techniques for eliminating distractors, managing time, and boosting accuracy on statistics multiple choice quizzes.
Cheat Sheet
Descriptive vs. Inferential Statistics -
Descriptive statistics summarize data using measures like mean, median, and standard deviation, while inferential statistics use samples to draw conclusions about populations (e.g., t-tests and confidence intervals). Remember the mnemonic "Describe to Derive" to recall that we describe data first before making inferences. These foundations appear frequently in statistics final exam questions and answers to test your conceptual clarity.
Key Probability Distributions -
The Normal distribution (N(μ,σ²)) and Student's t-distribution are staples in MCQ statistics tests; use the 68-95-99.7 rule for Normal curves and switch to t when σ is unknown and n<30. Write out Z = (X−μ)/σ and t = (X̄−μ)/(s/√n) to reinforce formulas from reputable university resources. Practicing with distribution tables will boost your accuracy on final exam stats practice.
Hypothesis Testing Framework -
Every statistics test question hinges on specifying H₀ and H, choosing α (often 0.05), computing a test statistic, and comparing the p-value. A quick trick: if p ≤ α, "reject H₀," otherwise "fail to reject H₀." Solidifying this flow helps on fast-paced statistics multiple choice sections.
Confidence Intervals Simplified -
Confidence intervals estimate population parameters; the formula X̄ ± z*(σ/√n) (or t* for small samples) gives a range with, say, 95% certainty. Memorize "Mean plus-minus Margin" to recall that margin = critical value times standard error. Frequent practice on statistics test questions will make CI calculations second nature.
Basics of Regression Analysis -
Linear regression uses ŷ = b₀ + bx to model relationships; compute slope b = Σ[(xᵢ−x̄)(yᵢ−ȳ)]/Σ(xᵢ−x̄)² and check R² for fit. A handy tip is "Slope from Covariance" to remember numerator ties to covariance and denominator to variance. These formula-driven points often appear in statistics quizzes to assess your applied understanding.
