Statistics And Probability Practice Exam

Question 1 Statistics And Probability

What is Bayes' theorem and what is its formula?

The standard error (SE) measures the variability of the sample mean across different samples: SE = σ / √n. As sample size increases, SE decreases, meaning sample means become more precise estimates of the population mean. It is derived from the CLT. A confounding variable is a third variable that influences both the independent and dependent variables, creating a spurious association. For example, ice cream sales and drowning deaths are correlated because both are influenced by temperature (the confounder). For two events A and B: P(A ∩ B) = P(A) × P(B|A). If A and B are independent, this simplifies to P(A ∩ B) = P(A) × P(B). This rule calculates the probability that both events occur. Bayes' theorem updates the probability of a hypothesis given new evidence: P(A|B) = P(B|A) × P(A) / P(B). It relates the posterior probability P(A|B) to the prior P(A), the likelihood P(B|A), and the marginal P(B).

Question 2 Statistics And Probability

How is Bayes' theorem applied in medical testing?

If a disease has prevalence P(D) = 0.01, test sensitivity P(+|D) = 0.95, and false positive rate P(+|D') = 0.05, then: P(D|+) = (0.95 × 0.01) / (0.95 × 0.01 + 0.05 × 0.99) ≈ 0.16. Even with a positive test, the probability of disease is only about 16%, illustrating the base rate fallacy. The CDF gives the probability that a random variable takes a value less than or equal to x: F(x) = P(X ≤ x). It is non-decreasing, ranges from 0 to 1, and is defined for both discrete and continuous random variables. The Poisson distribution models the number of events occurring in a fixed interval of time or space. PMF: P(X=k) = (λᵏ × e⁻λ) / k!, where λ is the average rate. Both mean and variance equal λ. It is used for rare events like arrivals, defects, or accidents. The normal (Gaussian) distribution is a symmetric, bell-shaped continuous distribution defined by mean μ and standard deviation σ. Its PDF is f(x) = (1/σ√(2π)) × e^(-(x-μ)²/(2σ²)). About 68% of data falls within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.

Question 3 Statistics And Probability

What is a Type I error?

Simple linear regression models the relationship between one independent variable (x) and one dependent variable (y) using a straight line: ŷ = b₀ + b₁x. The slope b₁ indicates the change in y for a one-unit increase in x, and b₀ is the y-intercept. Stratified sampling divides the population into homogeneous subgroups (strata) based on a characteristic, then randomly samples from each stratum. It ensures representation of all subgroups and can increase precision compared to simple random sampling when strata differ on the variable of interest. The base rate fallacy occurs when people ignore prior probabilities (base rates) and overweight new evidence. Bayes' theorem corrects this by explicitly incorporating the base rate. For rare conditions, even accurate tests can have high false positive rates relative to true positives. A Type I error (false positive) occurs when we reject a true null hypothesis. Its probability is denoted by α (the significance level). For example, concluding a drug is effective when it actually has no effect. Setting α = 0.05 means accepting a 5% risk of this error.

Question 4 Statistics And Probability

What is the geometric distribution?

The geometric distribution models the number of Bernoulli trials needed to get the first success. PMF: P(X=k) = (1-p)ᵏ⁻¹ × p. Its mean is 1/p and variance is (1-p)/p². It has the memoryless property. Residuals are the differences between observed and predicted values: eᵢ = yᵢ - ŷᵢ. Analyzing residual plots helps check regression assumptions. Randomly scattered residuals suggest a good fit, while patterns indicate issues like nonlinearity or heteroscedasticity. The normal (Gaussian) distribution is a symmetric, bell-shaped continuous distribution defined by mean μ and standard deviation σ. Its PDF is f(x) = (1/σ√(2π)) × e^(-(x-μ)²/(2σ²)). About 68% of data falls within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean. Key properties: Var(X) = E(X²) - [E(X)]². Var(aX + b) = a²Var(X). For independent variables: Var(X + Y) = Var(X) + Var(Y). Variance is always non-negative and equals zero only when X is constant.

Question 5 Statistics And Probability

What are residuals in regression analysis?

A z-test is used when the population standard deviation σ is known and the sample size is large (n ≥ 30). The test statistic is z = (x̄ - μ₀) / (σ / √n). It compares the sample mean to a hypothesized population mean under the normal distribution. The chi-square test of independence determines whether two categorical variables are related. It compares observed frequencies to expected frequencies: χ² = Σ (O - E)² / E, where E = (row total × column total) / grand total. A large χ² suggests the variables are not independent. Residuals are the differences between observed and predicted values: eᵢ = yᵢ - ŷᵢ. Analyzing residual plots helps check regression assumptions. Randomly scattered residuals suggest a good fit, while patterns indicate issues like nonlinearity or heteroscedasticity. Yes. For sample proportions, the CLT states that p̂ ~ N(p, √(p(1-p)/n)) when n is large enough (commonly np ≥ 10 and n(1-p) ≥ 10). This enables normal approximation for binomial proportions.

Question 6 Statistics And Probability

What are degrees of freedom in a chi-square test?

For a goodness-of-fit test: df = k - 1, where k is the number of categories. For a test of independence: df = (r - 1)(c - 1), where r is the number of rows and c is the number of columns in the contingency table. If the p-value ≤ α (significance level), reject H₀; the result is statistically significant. If the p-value > α, fail to reject H₀. For example, with α = 0.05, a p-value of 0.03 leads to rejection of H₀, while a p-value of 0.12 does not. The chi-square test of independence determines whether two categorical variables are related. It compares observed frequencies to expected frequencies: χ² = Σ (O - E)² / E, where E = (row total × column total) / grand total. A large χ² suggests the variables are not independent. The median is the middle value when data is sorted in order. For an even number of observations, it is the average of the two middle values. It is preferred over the mean when data is skewed or contains outliers, as it is more robust.

Question 7 Statistics And Probability

What is the Pearson correlation coefficient (r)?

A sampling distribution is the probability distribution of a statistic (like the mean) computed from all possible samples of a given size from a population. It describes how the statistic varies from sample to sample and is central to statistical inference. CI = x̄ ± z* × (σ / √n), where z* is the critical z-value for the desired confidence level (e.g., 1.96 for 95%). This requires knowing the population standard deviation or having a large sample size. Simple linear regression models the relationship between one independent variable (x) and one dependent variable (y) using a straight line: ŷ = b₀ + b₁x. The slope b₁ indicates the change in y for a one-unit increase in x, and b₀ is the y-intercept. Pearson's r measures the strength and direction of the linear relationship between two continuous variables. It ranges from -1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear relationship. Formula: r = Σ(xᵢ-x̄)(yᵢ-ȳ) / √[Σ(xᵢ-x̄)²Σ(yᵢ-ȳ)²].

Question 8 Statistics And Probability

What is the mode of a dataset?

The mode is the value that appears most frequently in a dataset. A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes). If no value repeats, the dataset has no mode. When n is large and p is small (typically n ≥ 20 and p ≤ 0.05), the binomial can be approximated by a Poisson with λ = np. This simplifies calculations when dealing with rare events over many trials. Larger sample sizes produce sampling distributions that are more tightly clustered around the population mean (smaller standard error) and more closely approximate a normal distribution. Even for heavily skewed populations, n ≥ 30 usually yields an approximately normal sampling distribution. A distribution is skewed when it is not symmetric. In a right-skewed (positive skew) distribution, the tail extends to the right and mean > median. In a left-skewed (negative skew) distribution, the tail extends to the left and mean < median.

Question 9 Statistics And Probability

When do you use a t-test instead of a z-test?

Use a t-test when the population standard deviation is unknown and must be estimated from the sample, especially with small samples (n < 30). The test statistic is t = (x̄ - μ₀) / (s / √n) with n-1 degrees of freedom. The t-distribution has heavier tails than the normal. When performing many statistical tests, the probability of at least one false positive increases. With 20 independent tests at α = 0.05, the expected number of false positives is 1. Corrections like Bonferroni (α/m) or Benjamini-Hochberg adjust for this inflated error rate. When n is large and p is small (typically n ≥ 20 and p ≤ 0.05), the binomial can be approximated by a Poisson with λ = np. This simplifies calculations when dealing with rare events over many trials. A distribution is skewed when it is not symmetric. In a right-skewed (positive skew) distribution, the tail extends to the right and mean > median. In a left-skewed (negative skew) distribution, the tail extends to the left and mean < median.

Question 10 Statistics And Probability

What is the range of a dataset?

A binomial experiment requires: (1) a fixed number of trials n, (2) each trial has exactly two outcomes (success/failure), (3) the probability of success p is constant across trials, and (4) trials are independent of each other. When n is large and p is small (typically n ≥ 20 and p ≤ 0.05), the binomial can be approximated by a Poisson with λ = np. This simplifies calculations when dealing with rare events over many trials. The range is the difference between the maximum and minimum values: Range = Max - Min. It is the simplest measure of dispersion but is highly sensitive to outliers and does not reflect how data is distributed between the extremes. Studies show a correlation between shoe size and reading ability in children. This does not mean bigger feet cause better reading. The confounding variable is age: older children have both larger feet and better reading skills. This illustrates why controlled experiments are needed.

Question 11 Statistics And Probability

What is convenience sampling and why is it problematic?

Convenience sampling selects participants who are easiest to reach (e.g., surveying people at a mall). It is prone to selection bias and produces samples that may not represent the population. Results from convenience samples generally cannot be generalized to the broader population. Statistical significance means the result is unlikely due to chance (p ≤ α), but the effect may be trivially small. Practical significance considers whether the effect size is large enough to be meaningful in context. Large samples can detect tiny, practically meaningless differences. The chi-square test of independence determines whether two categorical variables are related. It compares observed frequencies to expected frequencies: χ² = Σ (O - E)² / E, where E = (row total × column total) / grand total. A large χ² suggests the variables are not independent. A random variable is a numerical quantity whose value is determined by the outcome of a random experiment. It maps outcomes in a sample space to real numbers. Random variables can be discrete (countable values) or continuous (any value in an interval).

Question 12 Statistics And Probability

What does it mean for two events to be mutually exclusive?

Two events are mutually exclusive (disjoint) if they cannot occur at the same time: P(A ∩ B) = 0. For example, rolling a 3 and rolling a 5 on a single die are mutually exclusive. Mutually exclusive events are never independent (unless one has probability 0). The expected value E(X) is the long-run average value of a random variable. For discrete: E(X) = Σ xᵢ × P(xᵢ). For continuous: E(X) = ∫ x × f(x)dx. It is also called the population mean μ. The Poisson distribution models the number of events occurring in a fixed interval of time or space. PMF: P(X=k) = (λᵏ × e⁻λ) / k!, where λ is the average rate. Both mean and variance equal λ. It is used for rare events like arrivals, defects, or accidents. A PDF describes the probability distribution of a continuous random variable. The probability that X falls in an interval [a, b] is the area under the curve: P(a ≤ X ≤ b) = ∫ₐᵇ f(x)dx. The PDF itself can exceed 1, but the total area under it must equal 1.

Question 13 Statistics And Probability

How does sample size affect the sampling distribution according to the CLT?

The CLT states that the sampling distribution of the sample mean approaches a normal distribution as sample size n increases, regardless of the population's distribution. Specifically, x̄ ~ N(μ, σ/√n) for sufficiently large n (typically n ≥ 30). Bayes' theorem updates the probability of a hypothesis given new evidence: P(A|B) = P(B|A) × P(A) / P(B). It relates the posterior probability P(A|B) to the prior P(A), the likelihood P(B|A), and the marginal P(B). A data point is considered an outlier if it falls below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR. This method is used in box plots to identify unusually extreme values in a dataset. Larger sample sizes produce sampling distributions that are more tightly clustered around the population mean (smaller standard error) and more closely approximate a normal distribution. Even for heavily skewed populations, n ≥ 30 usually yields an approximately normal sampling distribution.

Question 14 Statistics And Probability

What is standard deviation and what does it measure?

Pearson's r measures the strength and direction of the linear relationship between two continuous variables. It ranges from -1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear relationship. Formula: r = Σ(xᵢ-x̄)(yᵢ-ȳ) / √[Σ(xᵢ-x̄)²Σ(yᵢ-ȳ)²]. The chi-square test of independence determines whether two categorical variables are related. It compares observed frequencies to expected frequencies: χ² = Σ (O - E)² / E, where E = (row total × column total) / grand total. A large χ² suggests the variables are not independent. The CLT is foundational because it justifies using normal-based methods (z-tests, confidence intervals) even when the population is not normally distributed. It enables inference about population parameters from sample data and underpins most parametric statistical tests. Standard deviation measures the average amount of dispersion or spread in a dataset relative to the mean. A low standard deviation indicates data points are close to the mean, while a high standard deviation indicates they are spread out. Formula: σ = √(Σ(xᵢ - x̄)² / n) for a population.

Question 15 Statistics And Probability

What does a p-value of 0.05 actually mean?

A random variable is a numerical quantity whose value is determined by the outcome of a random experiment. It maps outcomes in a sample space to real numbers. Random variables can be discrete (countable values) or continuous (any value in an interval). No. A small p-value indicates that the observed data is unlikely under H₀, but it does not prove H₁ is true. It could result from a real effect, a large sample size making a trivial effect significant, or chance. P-values do not measure the probability that H₀ is true. It means there is a 5% probability of obtaining a result as extreme as the observed one if the null hypothesis were true. It does not mean there is a 5% chance that H₀ is true, nor a 95% chance that H₁ is true. Quartiles divide sorted data into four equal parts. Q1 (25th percentile), Q2 (median, 50th percentile), and Q3 (75th percentile). The IQR is Q3 - Q1 and measures the spread of the middle 50% of data. It is robust to outliers.

Question 16 Statistics And Probability

What is spurious correlation?

Random assignment means participants are assigned to treatment or control groups by chance. It distributes confounding variables evenly across groups, allowing researchers to attribute differences in outcomes to the treatment. It is the key feature that distinguishes experiments from observational studies. Statistical significance means the result is unlikely due to chance (p ≤ α), but the effect may be trivially small. Practical significance considers whether the effect size is large enough to be meaningful in context. Large samples can detect tiny, practically meaningless differences. Correlation means two variables are associated or move together, but it does not imply that one causes the other. Causation means changes in one variable directly produce changes in another. Establishing causation requires controlled experiments or careful causal inference methods. A spurious correlation is a statistical relationship between two variables that is not caused by a direct link between them. It may arise from confounding variables, coincidence, or data dredging. Examples include correlations between unrelated trends that happen to move together over time.

Question 17 Statistics And Probability

What is the addition rule of probability?

Use a t-test when the population standard deviation is unknown and must be estimated from the sample, especially with small samples (n < 30). The test statistic is t = (x̄ - μ₀) / (s / √n) with n-1 degrees of freedom. The t-distribution has heavier tails than the normal. For any two events A and B: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). The subtraction of the intersection prevents double-counting. For mutually exclusive events, P(A ∩ B) = 0, so it simplifies to P(A ∪ B) = P(A) + P(B). The continuous uniform distribution has equal probability density over an interval [a, b]. PDF: f(x) = 1/(b-a) for a ≤ x ≤ b. Mean is (a+b)/2 and variance is (b-a)²/12. Every value in the interval is equally likely. The CLT states that the sampling distribution of the sample mean approaches a normal distribution as sample size n increases, regardless of the population's distribution. Specifically, x̄ ~ N(μ, σ/√n) for sufficiently large n (typically n ≥ 30).

Question 18 Statistics And Probability

What are the conditions for a binomial experiment?

When σ is unknown, use the t-distribution: CI = x̄ ± t* × (s / √n), where t* is the critical t-value with n-1 degrees of freedom and s is the sample standard deviation. For large n, t* approaches z*. The mode is the value that appears most frequently in a dataset. A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes). If no value repeats, the dataset has no mode. A binomial experiment requires: (1) a fixed number of trials n, (2) each trial has exactly two outcomes (success/failure), (3) the probability of success p is constant across trials, and (4) trials are independent of each other. If a disease has prevalence P(D) = 0.01, test sensitivity P(+|D) = 0.95, and false positive rate P(+|D') = 0.05, then: P(D|+) = (0.95 × 0.01) / (0.95 × 0.01 + 0.05 × 0.99) ≈ 0.16. Even with a positive test, the probability of disease is only about 16%, illustrating the base rate fallacy.

Question 19 Statistics And Probability

How does changing the confidence level affect the interval width?

Increasing the confidence level (e.g., from 95% to 99%) widens the interval because a larger critical value is used. There is a trade-off: higher confidence means a wider, less precise interval, while lower confidence gives a narrower but less reliable interval. The ecological fallacy occurs when conclusions about individuals are drawn from group-level (aggregate) data. For example, if countries with higher chocolate consumption have more Nobel laureates, it does not mean individuals who eat chocolate are more likely to win a Nobel Prize. Multiple regression extends simple regression to include two or more independent variables: ŷ = b₀ + b₁x₁ + b₂x₂ + ... + bₖxₖ. It allows analysis of how multiple predictors jointly influence the response variable while controlling for other variables. This test determines whether observed categorical data matches an expected distribution. The statistic is χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ. For example, testing whether a die is fair by comparing observed rolls to the expected uniform distribution.

Question 20 Statistics And Probability

What is a Type II error?

Sampling bias occurs when the sampling method systematically favors certain members of the population over others. This produces a non-representative sample and can lead to incorrect conclusions. Examples include voluntary response bias (only motivated people respond) and undercoverage. A random variable is a numerical quantity whose value is determined by the outcome of a random experiment. It maps outcomes in a sample space to real numbers. Random variables can be discrete (countable values) or continuous (any value in an interval). Statistical significance means the result is unlikely due to chance (p ≤ α), but the effect may be trivially small. Practical significance considers whether the effect size is large enough to be meaningful in context. Large samples can detect tiny, practically meaningless differences. A Type II error (false negative) occurs when we fail to reject a false null hypothesis. Its probability is denoted by β. For example, concluding a drug has no effect when it actually does. The probability of avoiding a Type II error is the power of the test: Power = 1 - β.

Question 21 Statistics And Probability

How do you detect outliers using the IQR method?

CI = x̄ ± z* × (σ / √n), where z* is the critical z-value for the desired confidence level (e.g., 1.96 for 95%). This requires knowing the population standard deviation or having a large sample size. Correlation means two variables are associated or move together, but it does not imply that one causes the other. Causation means changes in one variable directly produce changes in another. Establishing causation requires controlled experiments or careful causal inference methods. A data point is considered an outlier if it falls below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR. This method is used in box plots to identify unusually extreme values in a dataset. For two events A and B: P(A ∩ B) = P(A) × P(B|A). If A and B are independent, this simplifies to P(A ∩ B) = P(A) × P(B). This rule calculates the probability that both events occur.

Question 22 Statistics And Probability

What is the base rate fallacy and how does Bayes' theorem address it?

If a disease has prevalence P(D) = 0.01, test sensitivity P(+|D) = 0.95, and false positive rate P(+|D') = 0.05, then: P(D|+) = (0.95 × 0.01) / (0.95 × 0.01 + 0.05 × 0.99) ≈ 0.16. Even with a positive test, the probability of disease is only about 16%, illustrating the base rate fallacy. The base rate fallacy occurs when people ignore prior probabilities (base rates) and overweight new evidence. Bayes' theorem corrects this by explicitly incorporating the base rate. For rare conditions, even accurate tests can have high false positive rates relative to true positives. A distribution is skewed when it is not symmetric. In a right-skewed (positive skew) distribution, the tail extends to the right and mean > median. In a left-skewed (negative skew) distribution, the tail extends to the left and mean < median. The expected value E(X) is the long-run average value of a random variable. For discrete: E(X) = Σ xᵢ × P(xᵢ). For continuous: E(X) = ∫ x × f(x)dx. It is also called the population mean μ.

Question 23 Statistics And Probability

What is a controlled experiment?

Sampling bias occurs when the sampling method systematically favors certain members of the population over others. This produces a non-representative sample and can lead to incorrect conclusions. Examples include voluntary response bias (only motivated people respond) and undercoverage. If a disease has prevalence P(D) = 0.01, test sensitivity P(+|D) = 0.95, and false positive rate P(+|D') = 0.05, then: P(D|+) = (0.95 × 0.01) / (0.95 × 0.01 + 0.05 × 0.99) ≈ 0.16. Even with a positive test, the probability of disease is only about 16%, illustrating the base rate fallacy. A controlled experiment manipulates an independent variable while holding other variables constant to observe the effect on a dependent variable. It includes a treatment group and a control group. Random assignment helps ensure that differences in outcomes are due to the treatment, not confounding factors. Response bias occurs when respondents give inaccurate answers due to question wording, social desirability, leading questions, or misunderstanding. For example, people may underreport alcohol consumption. Careful survey design with neutral wording helps minimize response bias.

Question 24 Statistics And Probability

What is the empirical rule (68-95-99.7 rule)?

Multicollinearity occurs when independent variables in a regression model are highly correlated with each other. It inflates standard errors of coefficients, making them unreliable. Detection methods include the Variance Inflation Factor (VIF); VIF > 10 is often considered problematic. Simple linear regression models the relationship between one independent variable (x) and one dependent variable (y) using a straight line: ŷ = b₀ + b₁x. The slope b₁ indicates the change in y for a one-unit increase in x, and b₀ is the y-intercept. For a normal distribution: approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This rule provides quick estimates for the spread of normally distributed data. Stratified sampling divides the population into homogeneous subgroups (strata) based on a characteristic, then randomly samples from each stratum. It ensures representation of all subgroups and can increase precision compared to simple random sampling when strata differ on the variable of interest.

Question 25 Statistics And Probability

What is an independent two-sample t-test?

It compares the means of two independent groups to determine if they are significantly different. The test statistic is t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂). Assumptions include independence, normality, and (for pooled version) equal variances. The ecological fallacy occurs when conclusions about individuals are drawn from group-level (aggregate) data. For example, if countries with higher chocolate consumption have more Nobel laureates, it does not mean individuals who eat chocolate are more likely to win a Nobel Prize. The alternative hypothesis is the claim being tested, representing the effect or difference the researcher suspects exists. It can be one-tailed (μ > μ₀ or μ < μ₀) or two-tailed (μ ≠ μ₀). Rejecting H₀ provides support for H₁. A sampling distribution is the probability distribution of a statistic (like the mean) computed from all possible samples of a given size from a population. It describes how the statistic varies from sample to sample and is central to statistical inference.

Question 26 Statistics And Probability

What is a probability mass function (PMF)?

A PMF defines the probability distribution of a discrete random variable. It gives P(X = x) for each possible value x. The PMF must satisfy: all probabilities are between 0 and 1, and the sum of all probabilities equals 1. Use a t-test when the population standard deviation is unknown and must be estimated from the sample, especially with small samples (n < 30). The test statistic is t = (x̄ - μ₀) / (s / √n) with n-1 degrees of freedom. The t-distribution has heavier tails than the normal. The median is the middle value when data is sorted in order. For an even number of observations, it is the average of the two middle values. It is preferred over the mean when data is skewed or contains outliers, as it is more robust. Bayes' theorem updates the probability of a hypothesis given new evidence: P(A|B) = P(B|A) × P(A) / P(B). It relates the posterior probability P(A|B) to the prior P(A), the likelihood P(B|A), and the marginal P(B).

Question 27 Statistics And Probability

What is random assignment and why is it important?

A confidence interval is a range of values likely to contain the true population parameter. A 95% CI means that if we repeated the sampling process many times, approximately 95% of the constructed intervals would contain the true parameter. A controlled experiment manipulates an independent variable while holding other variables constant to observe the effect on a dependent variable. It includes a treatment group and a control group. Random assignment helps ensure that differences in outcomes are due to the treatment, not confounding factors. Random assignment means participants are assigned to treatment or control groups by chance. It distributes confounding variables evenly across groups, allowing researchers to attribute differences in outcomes to the treatment. It is the key feature that distinguishes experiments from observational studies. In systematic sampling, every kth element is selected from a list after a random starting point. For example, selecting every 10th name from an alphabetical list. It is simple to implement but can produce biased results if the list has a periodic pattern.

Question 28 Statistics And Probability

What is the chi-square test of independence?

When performing many statistical tests, the probability of at least one false positive increases. With 20 independent tests at α = 0.05, the expected number of false positives is 1. Corrections like Bonferroni (α/m) or Benjamini-Hochberg adjust for this inflated error rate. Multicollinearity occurs when independent variables in a regression model are highly correlated with each other. It inflates standard errors of coefficients, making them unreliable. Detection methods include the Variance Inflation Factor (VIF); VIF > 10 is often considered problematic. Hypothesis testing is a statistical method for making decisions about population parameters based on sample data. It involves stating a null hypothesis (H₀) and an alternative hypothesis (H₁), then using sample evidence to decide whether to reject H₀ at a specified significance level. The chi-square test of independence determines whether two categorical variables are related. It compares observed frequencies to expected frequencies: χ² = Σ (O - E)² / E, where E = (row total × column total) / grand total. A large χ² suggests the variables are not independent.

Question 29 Statistics And Probability

What is R-squared (coefficient of determination)?

R² measures the proportion of variance in the dependent variable explained by the independent variable(s). It ranges from 0 to 1; an R² of 0.75 means 75% of the variance in y is explained by the model. For simple regression, R² = r². Adjusted R² modifies R² to account for the number of predictors in a model. Unlike R², it can decrease when irrelevant variables are added. Formula: R²_adj = 1 - [(1-R²)(n-1)/(n-k-1)], where k is the number of predictors. It is preferred for comparing models with different numbers of variables. The Poisson distribution models the number of events occurring in a fixed interval of time or space. PMF: P(X=k) = (λᵏ × e⁻λ) / k!, where λ is the average rate. Both mean and variance equal λ. It is used for rare events like arrivals, defects, or accidents. Sampling bias occurs when the sampling method systematically favors certain members of the population over others. This produces a non-representative sample and can lead to incorrect conclusions. Examples include voluntary response bias (only motivated people respond) and undercoverage.

Question 30 Statistics And Probability

What is the null hypothesis (H₀)?

CI = p̂ ± z* × √(p̂(1-p̂)/n), where p̂ is the sample proportion and z* is the critical z-value. This assumes np̂ ≥ 10 and n(1-p̂) ≥ 10 for the normal approximation to be valid. Conditional probability is the probability of event A given that event B has occurred: P(A|B) = P(A ∩ B) / P(B), where P(B) > 0. It updates the probability of A based on new information that B is true. The null hypothesis is the default assumption that there is no effect, no difference, or no relationship. It typically states that a parameter equals a specific value (e.g., μ = 0). It is assumed true until sufficient evidence is found to reject it. The base rate fallacy occurs when people ignore prior probabilities (base rates) and overweight new evidence. Bayes' theorem corrects this by explicitly incorporating the base rate. For rare conditions, even accurate tests can have high false positive rates relative to true positives.

Question 31 Statistics And Probability

What is the ecological fallacy in correlation studies?

The standard normal distribution has mean μ = 0 and standard deviation σ = 1. Any normal variable X can be standardized to Z using Z = (X - μ) / σ. Z-tables or software are used to find probabilities for the standard normal distribution. CI = x̄ ± z* × (σ / √n), where z* is the critical z-value for the desired confidence level (e.g., 1.96 for 95%). This requires knowing the population standard deviation or having a large sample size. The ecological fallacy occurs when conclusions about individuals are drawn from group-level (aggregate) data. For example, if countries with higher chocolate consumption have more Nobel laureates, it does not mean individuals who eat chocolate are more likely to win a Nobel Prize. A controlled experiment manipulates an independent variable while holding other variables constant to observe the effect on a dependent variable. It includes a treatment group and a control group. Random assignment helps ensure that differences in outcomes are due to the treatment, not confounding factors.

Question 32 Statistics And Probability

What is a p-value?

A p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed from sample data, assuming the null hypothesis is true. It quantifies the strength of evidence against H₀. A smaller p-value indicates stronger evidence against H₀. A Type I error (false positive) occurs when we reject a true null hypothesis. Its probability is denoted by α (the significance level). For example, concluding a drug is effective when it actually has no effect. Setting α = 0.05 means accepting a 5% risk of this error. CI = x̄ ± z* × (σ / √n), where z* is the critical z-value for the desired confidence level (e.g., 1.96 for 95%). This requires knowing the population standard deviation or having a large sample size. It compares the means of two independent groups to determine if they are significantly different. The test statistic is t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂). Assumptions include independence, normality, and (for pooled version) equal variances.

Question 33 Statistics And Probability

What is conditional probability?

Yes. For sample proportions, the CLT states that p̂ ~ N(p, √(p(1-p)/n)) when n is large enough (commonly np ≥ 10 and n(1-p) ≥ 10). This enables normal approximation for binomial proportions. Conditional probability is the probability of event A given that event B has occurred: P(A|B) = P(A ∩ B) / P(B), where P(B) > 0. It updates the probability of A based on new information that B is true. The geometric distribution models the number of Bernoulli trials needed to get the first success. PMF: P(X=k) = (1-p)ᵏ⁻¹ × p. Its mean is 1/p and variance is (1-p)/p². It has the memoryless property. A sampling distribution is the probability distribution of a statistic (like the mean) computed from all possible samples of a given size from a population. It describes how the statistic varies from sample to sample and is central to statistical inference.

Question 34 Statistics And Probability

Does the CLT apply to proportions as well as means?

A data point is considered an outlier if it falls below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR. This method is used in box plots to identify unusually extreme values in a dataset. The CDF gives the probability that a random variable takes a value less than or equal to x: F(x) = P(X ≤ x). It is non-decreasing, ranges from 0 to 1, and is defined for both discrete and continuous random variables. If a disease has prevalence P(D) = 0.01, test sensitivity P(+|D) = 0.95, and false positive rate P(+|D') = 0.05, then: P(D|+) = (0.95 × 0.01) / (0.95 × 0.01 + 0.05 × 0.99) ≈ 0.16. Even with a positive test, the probability of disease is only about 16%, illustrating the base rate fallacy. Yes. For sample proportions, the CLT states that p̂ ~ N(p, √(p(1-p)/n)) when n is large enough (commonly np ≥ 10 and n(1-p) ≥ 10). This enables normal approximation for binomial proportions.

Question 35 Statistics And Probability

What is a likelihood ratio in Bayesian analysis?

The standard error (SE) measures the variability of the sample mean across different samples: SE = σ / √n. As sample size increases, SE decreases, meaning sample means become more precise estimates of the population mean. It is derived from the CLT. The likelihood ratio is P(B|A) / P(B|A'), comparing how likely the evidence is under the hypothesis versus the alternative. Multiplying the prior odds by the likelihood ratio gives the posterior odds: Posterior odds = Prior odds × Likelihood ratio. Increasing sample size decreases the standard error (σ/√n), which narrows the confidence interval. This provides a more precise estimate of the population parameter. Quadrupling the sample size halves the width of the interval. The CLT is foundational because it justifies using normal-based methods (z-tests, confidence intervals) even when the population is not normally distributed. It enables inference about population parameters from sample data and underpins most parametric statistical tests.

Question 36 Statistics And Probability

What is multicollinearity in regression?

When σ is unknown, use the t-distribution: CI = x̄ ± t* × (s / √n), where t* is the critical t-value with n-1 degrees of freedom and s is the sample standard deviation. For large n, t* approaches z*. Larger sample sizes produce sampling distributions that are more tightly clustered around the population mean (smaller standard error) and more closely approximate a normal distribution. Even for heavily skewed populations, n ≥ 30 usually yields an approximately normal sampling distribution. Residuals are the differences between observed and predicted values: eᵢ = yᵢ - ŷᵢ. Analyzing residual plots helps check regression assumptions. Randomly scattered residuals suggest a good fit, while patterns indicate issues like nonlinearity or heteroscedasticity. Multicollinearity occurs when independent variables in a regression model are highly correlated with each other. It inflates standard errors of coefficients, making them unreliable. Detection methods include the Variance Inflation Factor (VIF); VIF > 10 is often considered problematic.

Question 37 Statistics And Probability

What is simple random sampling?

When n is large and p is small (typically n ≥ 20 and p ≤ 0.05), the binomial can be approximated by a Poisson with λ = np. This simplifies calculations when dealing with rare events over many trials. The geometric distribution models the number of Bernoulli trials needed to get the first success. PMF: P(X=k) = (1-p)ᵏ⁻¹ × p. Its mean is 1/p and variance is (1-p)/p². It has the memoryless property. Simple random sampling gives every member of the population an equal probability of being selected. It can be implemented using random number generators. It is the gold standard for unbiased sampling but may be impractical for large or geographically dispersed populations. Simple linear regression models the relationship between one independent variable (x) and one dependent variable (y) using a straight line: ŷ = b₀ + b₁x. The slope b₁ indicates the change in y for a one-unit increase in x, and b₀ is the y-intercept.

Question 38 Statistics And Probability

What are the properties of variance for random variables?

Key properties: Var(X) = E(X²) - [E(X)]². Var(aX + b) = a²Var(X). For independent variables: Var(X + Y) = Var(X) + Var(Y). Variance is always non-negative and equals zero only when X is constant. The mode is the value that appears most frequently in a dataset. A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes). If no value repeats, the dataset has no mode. R² measures the proportion of variance in the dependent variable explained by the independent variable(s). It ranges from 0 to 1; an R² of 0.75 means 75% of the variance in y is explained by the model. For simple regression, R² = r². A p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed from sample data, assuming the null hypothesis is true. It quantifies the strength of evidence against H₀. A smaller p-value indicates stronger evidence against H₀.

Question 39 Statistics And Probability

What is the Central Limit Theorem (CLT)?

The normal (Gaussian) distribution is a symmetric, bell-shaped continuous distribution defined by mean μ and standard deviation σ. Its PDF is f(x) = (1/σ√(2π)) × e^(-(x-μ)²/(2σ²)). About 68% of data falls within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean. A Type II error (false negative) occurs when we fail to reject a false null hypothesis. Its probability is denoted by β. For example, concluding a drug has no effect when it actually does. The probability of avoiding a Type II error is the power of the test: Power = 1 - β. For two events A and B: P(A ∩ B) = P(A) × P(B|A). If A and B are independent, this simplifies to P(A ∩ B) = P(A) × P(B). This rule calculates the probability that both events occur. The CLT states that the sampling distribution of the sample mean approaches a normal distribution as sample size n increases, regardless of the population's distribution. Specifically, x̄ ~ N(μ, σ/√n) for sufficiently large n (typically n ≥ 30).

Question 40 Statistics And Probability

What is a skewed distribution?

A distribution is skewed when it is not symmetric. In a right-skewed (positive skew) distribution, the tail extends to the right and mean > median. In a left-skewed (negative skew) distribution, the tail extends to the left and mean < median. A z-test is used when the population standard deviation σ is known and the sample size is large (n ≥ 30). The test statistic is z = (x̄ - μ₀) / (σ / √n). It compares the sample mean to a hypothesized population mean under the normal distribution. Quartiles divide sorted data into four equal parts. Q1 (25th percentile), Q2 (median, 50th percentile), and Q3 (75th percentile). The IQR is Q3 - Q1 and measures the spread of the middle 50% of data. It is robust to outliers. A sampling distribution is the probability distribution of a statistic (like the mean) computed from all possible samples of a given size from a population. It describes how the statistic varies from sample to sample and is central to statistical inference.

Question 41 Statistics And Probability

What is the chi-square goodness-of-fit test?

This test determines whether observed categorical data matches an expected distribution. The statistic is χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ. For example, testing whether a die is fair by comparing observed rolls to the expected uniform distribution. Convenience sampling selects participants who are easiest to reach (e.g., surveying people at a mall). It is prone to selection bias and produces samples that may not represent the population. Results from convenience samples generally cannot be generalized to the broader population. Variance is the average of the squared differences from the mean: σ² = Σ(xᵢ - x̄)² / n. Standard deviation is the square root of variance. Variance is expressed in squared units of the original data, making standard deviation more interpretable. Correlation means two variables are associated or move together, but it does not imply that one causes the other. Causation means changes in one variable directly produce changes in another. Establishing causation requires controlled experiments or careful causal inference methods.

Question 42 Statistics And Probability

What is the standard error of the mean?

The standard error (SE) measures the variability of the sample mean across different samples: SE = σ / √n. As sample size increases, SE decreases, meaning sample means become more precise estimates of the population mean. It is derived from the CLT. Statistical power is the probability of correctly rejecting a false null hypothesis: Power = 1 - β. It increases with larger sample sizes, larger effect sizes, higher significance levels (α), and lower variability. A common target is power ≥ 0.80. The expected value E(X) is the long-run average value of a random variable. For discrete: E(X) = Σ xᵢ × P(xᵢ). For continuous: E(X) = ∫ x × f(x)dx. It is also called the population mean μ. The prior probability P(A) is the initial belief about the probability of an event before new evidence. The posterior probability P(A|B) is the updated probability after considering new evidence B. Bayes' theorem provides the mathematical framework for this update.

Question 43 Statistics And Probability

What is the complement rule of probability?

Conditional probability is the probability of event A given that event B has occurred: P(A|B) = P(A ∩ B) / P(B), where P(B) > 0. It updates the probability of A based on new information that B is true. Studies show a correlation between shoe size and reading ability in children. This does not mean bigger feet cause better reading. The confounding variable is age: older children have both larger feet and better reading skills. This illustrates why controlled experiments are needed. The complement of event A (denoted A' or Aᶜ) includes all outcomes not in A. The rule states: P(A') = 1 - P(A). This is often useful when calculating the probability of "at least one" event by computing 1 - P(none). The significance level α is the threshold for rejecting H₀, representing the maximum acceptable probability of a Type I error. Common choices are 0.05, 0.01, and 0.10. It should be set before collecting data, based on the consequences of a false positive.

Question 44 Statistics And Probability

What is the significance level (α) and how is it chosen?

The median is the middle value when data is sorted in order. For an even number of observations, it is the average of the two middle values. It is preferred over the mean when data is skewed or contains outliers, as it is more robust. Increasing sample size decreases the standard error (σ/√n), which narrows the confidence interval. This provides a more precise estimate of the population parameter. Quadrupling the sample size halves the width of the interval. In an observational study, the researcher observes and measures variables without manipulating them. Unlike experiments, observational studies cannot establish causation because there is no random assignment to eliminate confounders. They are used when experiments are impractical or unethical. The significance level α is the threshold for rejecting H₀, representing the maximum acceptable probability of a Type I error. Common choices are 0.05, 0.01, and 0.10. It should be set before collecting data, based on the consequences of a false positive.

Question 45 Statistics And Probability

How do you interpret a p-value in hypothesis testing?

If the p-value ≤ α (significance level), reject H₀; the result is statistically significant. If the p-value > α, fail to reject H₀. For example, with α = 0.05, a p-value of 0.03 leads to rejection of H₀, while a p-value of 0.12 does not. A z-test is used when the population standard deviation σ is known and the sample size is large (n ≥ 30). The test statistic is z = (x̄ - μ₀) / (σ / √n). It compares the sample mean to a hypothesized population mean under the normal distribution. Conditional probability is the probability of event A given that event B has occurred: P(A|B) = P(A ∩ B) / P(B), where P(B) > 0. It updates the probability of A based on new information that B is true. The standard error (SE) measures the variability of the sample mean across different samples: SE = σ / √n. As sample size increases, SE decreases, meaning sample means become more precise estimates of the population mean. It is derived from the CLT.

Question 46 Statistics And Probability

What is a confidence interval?

The base rate fallacy occurs when people ignore prior probabilities (base rates) and overweight new evidence. Bayes' theorem corrects this by explicitly incorporating the base rate. For rare conditions, even accurate tests can have high false positive rates relative to true positives. A PMF defines the probability distribution of a discrete random variable. It gives P(X = x) for each possible value x. The PMF must satisfy: all probabilities are between 0 and 1, and the sum of all probabilities equals 1. A confidence interval is a range of values likely to contain the true population parameter. A 95% CI means that if we repeated the sampling process many times, approximately 95% of the constructed intervals would contain the true parameter. The complement of event A (denoted A' or Aᶜ) includes all outcomes not in A. The rule states: P(A') = 1 - P(A). This is often useful when calculating the probability of "at least one" event by computing 1 - P(none).

Question 47 Statistics And Probability

What is the least squares method in regression?

Studies show a correlation between shoe size and reading ability in children. This does not mean bigger feet cause better reading. The confounding variable is age: older children have both larger feet and better reading skills. This illustrates why controlled experiments are needed. In an observational study, the researcher observes and measures variables without manipulating them. Unlike experiments, observational studies cannot establish causation because there is no random assignment to eliminate confounders. They are used when experiments are impractical or unethical. The least squares method finds the regression line that minimizes the sum of squared residuals: Σ(yᵢ - ŷᵢ)². The slope is b₁ = Σ(xᵢ-x̄)(yᵢ-ȳ) / Σ(xᵢ-x̄)² and the intercept is b₀ = ȳ - b₁x̄. A placebo is an inactive treatment (like a sugar pill) given to the control group. The placebo effect is the phenomenon where participants improve simply because they believe they are receiving treatment. Using placebos helps researchers isolate the actual effect of the treatment.

Question 48 Statistics And Probability

What is a placebo and the placebo effect?

The Poisson distribution models the number of events occurring in a fixed interval of time or space. PMF: P(X=k) = (λᵏ × e⁻λ) / k!, where λ is the average rate. Both mean and variance equal λ. It is used for rare events like arrivals, defects, or accidents. The chi-square test of independence determines whether two categorical variables are related. It compares observed frequencies to expected frequencies: χ² = Σ (O - E)² / E, where E = (row total × column total) / grand total. A large χ² suggests the variables are not independent. The mode is the value that appears most frequently in a dataset. A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes). If no value repeats, the dataset has no mode. A placebo is an inactive treatment (like a sugar pill) given to the control group. The placebo effect is the phenomenon where participants improve simply because they believe they are receiving treatment. Using placebos helps researchers isolate the actual effect of the treatment.

Question 49 Statistics And Probability

What is the exponential distribution?

Yes. For sample proportions, the CLT states that p̂ ~ N(p, √(p(1-p)/n)) when n is large enough (commonly np ≥ 10 and n(1-p) ≥ 10). This enables normal approximation for binomial proportions. If B₁, B₂, ..., Bₙ form a partition of the sample space, then for any event A: P(A) = Σ P(A|Bᵢ) × P(Bᵢ). This allows computing P(A) by breaking it down across all possible scenarios. The exponential distribution models the time between events in a Poisson process. PDF: f(x) = λe⁻λˣ for x ≥ 0. Mean is 1/λ and variance is 1/λ². It has the memoryless property: P(X > s+t | X > s) = P(X > t). Variance is the average of the squared differences from the mean: σ² = Σ(xᵢ - x̄)² / n. Standard deviation is the square root of variance. Variance is expressed in squared units of the original data, making standard deviation more interpretable.

Question 50 Statistics And Probability

What is the standard normal distribution?

For a goodness-of-fit test: df = k - 1, where k is the number of categories. For a test of independence: df = (r - 1)(c - 1), where r is the number of rows and c is the number of columns in the contingency table. The standard normal distribution has mean μ = 0 and standard deviation σ = 1. Any normal variable X can be standardized to Z using Z = (X - μ) / σ. Z-tables or software are used to find probabilities for the standard normal distribution. A z-test is used when the population standard deviation σ is known and the sample size is large (n ≥ 30). The test statistic is z = (x̄ - μ₀) / (σ / √n). It compares the sample mean to a hypothesized population mean under the normal distribution. If the p-value ≤ α (significance level), reject H₀; the result is statistically significant. If the p-value > α, fail to reject H₀. For example, with α = 0.05, a p-value of 0.03 leads to rejection of H₀, while a p-value of 0.12 does not.