Statistical Analysis and Hypothesis Testing Techniques

What is the normal approximation for the probability of four or fewer defective spark plugs in a sample of 200 with a defect rate of 1%?

Use the normal approximation: mean = np = 2, variance = np(1-p) = 1.98, then calculate P(X ≤ 4) using the z-score.

How do you calculate the area between 15 and 40 seconds in an exponential distribution with a mean of 20 seconds?

Use the cumulative distribution function (CDF) of the exponential distribution: P(15 < X < 40) = F(40) - F(15).

What is a p-value in hypothesis testing?

A p-value indicates the probability of observing the test results under the null hypothesis.

What is the null hypothesis in hypothesis testing?

The null hypothesis is a statement that there is no effect or no difference, and it is what we seek to test against.

What is the chi-square test used for in the context of a cross-tabulation table?

To assess whether there is a significant association between two categorical variables.

What are the null and alternative hypotheses for testing if the population mean expenditure on cat food is less than $100?

Null hypothesis (H0): μ ≥ 100; Alternative hypothesis (H1): μ < 100.

What are the null and alternative hypotheses for testing if the population mean expenditures on cat food in January and February are different?

Null hypothesis (H0): μ1 = μ2; Alternative hypothesis (H1): μ1 ≠ μ2.

What are the null and alternative hypotheses for testing if the population mean expenditures in Broward and Dade counties are different?

Null hypothesis (H0): μBroward = μDade; Alternative hypothesis (H1): μBroward ≠ μDade.

What is the primary purpose of principal component analysis (PCA)?

To reduce the dimensionality of data while preserving as much variance as possible.

How should eigenvalues in PCA be interpreted?

Eigenvalues indicate the amount of variance captured by each principal component.

What are principal component loadings?

Principal component loadings represent the correlation between the original variables and the principal components.

What is the purpose of rotating component loadings in PCA?

To achieve a simpler and more interpretable structure of the principal components.

How can PCA be used in conjunction with cluster analysis?

PCA can reduce dimensions before clustering, improving the efficiency and effectiveness of the clustering process.

How can PCA be used in conjunction with regression?

PCA can help in multicollinearity situations by transforming correlated predictors into uncorrelated principal components.

What are the eigenvalues and lead eigenvector of a correlation matrix?

The eigenvalues indicate the amount of variance explained by each component, while the lead eigenvector represents the direction of the first principal component.

How do you interpret the rotated factor loadings for scale items on components?

Interpretation involves identifying which scale items load highly on each component to define it, and noting items that do not load significantly on any component.

What is the difference between exploratory factor analysis (EFA) and principal component analysis (PCA)?

EFA aims to identify underlying relationships between variables, while PCA focuses on reducing dimensionality by transforming variables into principal components.

How does confirmatory factor analysis (CFA) differ from exploratory factor analysis (EFA)?

CFA tests a hypothesized model of factor structure, while EFA explores data to find potential structures without prior assumptions.

What parameters are estimated in a confirmatory factor analysis with multiple correlated factors?

Parameters include factor loadings, variances, and covariances, with the objective function typically being the minimization of the difference between the observed and predicted covariance matrices.

What are the steps involved in applying k-means clustering to a data set?

1. Choose the number of clusters (k). 2. Initialize cluster centroids. 3. Assign observations to the nearest centroid. 4. Update centroids based on assigned observations. 5. Repeat steps 3-4 until convergence.

Describe the steps for applying complete-linkage hierarchical clustering.

1. Calculate the distance matrix. 2. Identify the closest pair of clusters. 3. Merge these clusters. 4. Update the distance matrix. 5. Repeat until all observations are clustered.

What is the difference between hierarchical and non-hierarchical clustering methods?

Hierarchical clustering builds a tree of clusters, while non-hierarchical methods like k-means partition data into a fixed number of clusters without a hierarchy.

What are the two major categories of hierarchical clustering procedures?

The two major categories are agglomerative (bottom-up) and divisive (top-down) clustering methods.

How can Ward's method output help in choosing the number of clusters?

1. Examine the agglomeration coefficients to find a significant increase indicating the optimal number of clusters. 2. Use dendrograms to visually assess where to cut for the desired number of clusters.

What is the difference between single-linkage and complete-linkage cluster analysis?

Single-linkage uses the minimum distance between points in different clusters, while complete-linkage uses the maximum distance.

When should clustering variables be standardized before K-means clustering?

Standardization is prudent when variables are on different scales or units to ensure that all variables contribute equally to the distance calculations.

How do you compute the Ward's method agglomeration coefficient for cluster mergers?

Calculate the increase in the total within-cluster variance when merging clusters and select the merger that results in the smallest increase.

What is the squared Euclidean distance?

It is the sum of the squared differences between corresponding coordinates of two points, often used in clustering to measure similarity.

How do you interpret a matrix of distances between observations in clustering?

The matrix shows the squared distances, where lower values indicate closer observations, helping to determine cluster formation.

What is the significance of the first component in factor analysis?

The first component explains the largest amount of variance in the data, making it crucial for understanding the underlying structure.

How can you identify scale items that do not seem relevant to any component?

Items with low loadings across all components can be considered irrelevant to the factor structure identified.

What is the role of cluster centroids in K-means clustering?

Centroids represent the average position of all points in a cluster and are used to assign new observations to the nearest cluster.

What is the purpose of using a distance matrix in hierarchical clustering?

The distance matrix quantifies the similarity or dissimilarity between observations, guiding the clustering process.

What does a high loading on a component indicate?

A high loading indicates a strong relationship between the item and the component, suggesting the item is a key indicator of that component.

What is the objective of exploratory factor analysis?

The objective is to uncover the underlying structure of data by identifying latent variables that explain observed correlations.

What is the significance of the agglomeration coefficient in hierarchical clustering?

The agglomeration coefficient measures the increase in variance when clusters are merged, helping to determine the best clustering solution.

What is the first step in performing an iteration of K-means clustering?

Compute the current centroids (cluster centers) based on the assigned objects.

How do you determine the distances in K-means clustering?

Calculate the distances from each object to the centroids.

What is the purpose of reassigning objects in K-means clustering?

To assign each object to its nearest cluster center based on the computed distances.

What is multicollinearity in the context of regression analysis?

Multicollinearity refers to the situation where two or more predictor variables in a regression model are highly correlated, leading to unreliable coefficient estimates.

What are the potential consequences of multicollinearity?

It can lead to inflated standard errors, making it difficult to determine the effect of each predictor on the dependent variable, and can reduce the model's predictive power.

Why is model/variable selection important in regression analysis?

It helps to identify the most relevant predictors, improves model interpretability, and enhances prediction accuracy.

Compare two regression model/variable selection approaches.

Stepwise selection adds or removes predictors based on statistical criteria, while Lasso regression penalizes the absolute size of coefficients to shrink some to zero, effectively selecting variables.

What does the bivariate (Pearson) correlation measure indicate?

It measures the strength and direction of the linear relationship between two variables x and y.

How does partial correlation differ from bivariate correlation?

Partial correlation measures the relationship between two variables while controlling for the effect of one or more additional variables.

What is the interpretation of a slope coefficient in regression?

It indicates the expected change in the dependent variable for a one-unit increase in the predictor variable.

What is the range of predictors issue in regression analysis?

It refers to the problem that arises when the range of the predictor variable does not adequately cover the range of the dependent variable, potentially leading to biased estimates.

What is the interpretation of the coefficient (b1) for variable x1 in OLS regression?

It represents the change in the dependent variable for a one-unit increase in x1, holding all other variables constant.

How does the interpretation of the coefficient change in logistic regression?

In logistic regression, the coefficient represents the change in the log-odds of the dependent variable for a one-unit increase in x1.

What is the product-moment correlation coefficient?

It is a measure that quantifies the degree of linear relationship between two variables, ranging from -1 to 1.

How do you compute the product-moment correlation coefficient?

Calculate the covariance of the two variables divided by the product of their standard deviations.

What is the null hypothesis when testing the population correlation coefficient?

The null hypothesis states that the population correlation coefficient is zero (H0: ρ = 0).

What is the significance of a correlation of 0.83 between awareness and attitude?

It indicates a strong positive relationship between awareness and attitude.

How do you compute partial correlation between two variables adjusting for a third?

Use the formula for partial correlation which involves the correlation coefficients among the three variables involved.

What is the clustering index in complete-linkage clustering?

It quantifies the distance between clusters based on the maximum distance between any two objects in the clusters.

What is the significance of showing all work in clustering and correlation calculations?

It ensures transparency and allows for verification of the steps taken to arrive at the results.

What are the null and alternative hypotheses for the overall regression model predicting expenditure on Nike products?

Null hypothesis (H0): At least one of the independent variables (X1, X2, X3, X4) does not significantly predict expenditure (Y). Alternative hypothesis (H1): At least one of the independent variables significantly predicts expenditure.

What is the significance level assumed in the regression model analysis?

The significance level assumed is α = 0.05.

What are two common reasons for not using OLS regression when the dependent variable is binary?

1. OLS regression can predict probabilities outside the [0, 1] range. 2. The assumptions of OLS (e.g., homoscedasticity) are violated with binary outcomes.

What is the objective function optimized when fitting a logistic regression model?

The objective function is the likelihood function, which is maximized to estimate the coefficients.

What is one advantage and one disadvantage of logistic regression compared to discriminant analysis for two-group classification?

Advantage: Logistic regression does not assume normality of predictors. Disadvantage: Logistic regression may require larger sample sizes to achieve stable estimates.

What is l1-regularized logistic regression and its purpose?

L1-regularized logistic regression adds a penalty equal to the absolute value of the coefficients to the loss function, promoting sparsity in the model by reducing some coefficients to zero.

What is 'best subsets' logistic regression? How does it compare with l1-regularized logistic regression?

'Best subsets' logistic regression selects the best combination of predictors based on a criterion (e.g., AIC), while l1-regularization shrinks coefficients to achieve a simpler model.

Given coefficients b0 = -5.0074, b1 = 0.4313, b2 = 0.2095, b3 = -0.4634, b4 = 0.2549, and measurements x1 = 7, x2 = 7, x3 = 3, x4 = 7, how do you compute the probability of brand usage?

Calculate the linear predictor: z = b0 + b1x1 + b2x2 + b3x3 + b4x4, then use the logistic function P(Y=1) = 1 / (1 + e^(-z)).

What is the interpretation of exp(b3) in logistic regression?

exp(b3) represents the odds ratio for a one-unit increase in predictor X3, indicating how the odds of the outcome change.

What is the interpretation of Wilks' lambda in two-group discriminant analysis?

Wilks' lambda measures the ratio of the error variance to the total variance, with lower values indicating better discrimination between groups.

What is the relationship between Wilks' lambda and the eigenvalue in two-group discriminant analysis SPSS output?

Wilks' lambda is related to the eigenvalue as it reflects the proportion of variance explained by the discriminant function; lower Wilks' lambda corresponds to higher eigenvalues.

What does a number in the structure matrix of the discriminant function output represent?

It indicates the correlation between each predictor and the discriminant function, showing the contribution of each predictor to the discrimination.

How are standardized and unstandardized discriminant function coefficients used in analyses?

Standardized coefficients allow comparison of predictor importance, while unstandardized coefficients are used to calculate the discriminant score.

What is the null hypothesis of the chi-square test based on Wilks' lambda in two-group discriminant analysis?

The null hypothesis states that the group means are equal across the predictors.

What are important changes in discriminant analysis when moving from two groups to three groups?

Increased complexity in modeling, the need for more discriminant functions, and potential challenges in interpreting results.

What is the difference between one-mode and two-mode (bipartite) networks?

One-mode networks consist of a single type of node, while two-mode networks consist of two different types of nodes with connections between them.

What is the difference between directed and undirected networks?

Directed networks have edges with a specific direction (from one node to another), while undirected networks have edges without direction.

What is a signed network?

A signed network includes edges that can have positive or negative signs, indicating the nature of the relationship between nodes.

What is the typical objective when partitioning a signed network?

The objective is to maximize the number of positive relationships within groups while minimizing negative relationships between groups.

What is degree centrality for an undirected network? How does eigenvector centrality for a directed network differ?

Degree centrality counts the number of direct connections a node has, while eigenvector centrality considers the influence of a node's connections.

How do you compute in-degree and out-degree centrality for each vertex in a directed network?

In-degree centrality counts incoming edges to a vertex, while out-degree centrality counts outgoing edges from a vertex.

What does the first column of the t-table represent?

Degrees of freedom.

What does the first row of the t-table correspond to?

Areas in one of the tails.

What is the critical t value for 1 degree of freedom at a significance level of 0.05?

6.3138.

What is the critical t value for 10 degrees of freedom at a significance level of 0.01?

2.7638.

What is the critical t value for 30 degrees of freedom at a significance level of 0.005?

2.7500.

What is the critical value for F tables at a significance level of 0.05 for 1 numerator and 1 denominator degree of freedom?

161.45.

What is the critical F value for 2 numerator and 2 denominator degrees of freedom at a significance level of 0.01?

19.00.

What is the critical F value for 5 numerator and 5 denominator degrees of freedom at a significance level of 0.05?

4.88.

What is the critical t value for 4 degrees of freedom at a significance level of 0.025?

2.7764.

What is the critical t value for 12 degrees of freedom at a significance level of 0.1?

1.3562.

What is the critical t value for 20 degrees of freedom at a significance level of 0.2?

0.8600.

What is the critical F value for 10 numerator and 20 denominator degrees of freedom at a significance level of 0.05?

3.33.

What is the critical F value for 5 numerator and 10 denominator degrees of freedom at a significance level of 0.01?

4.96.

What is the critical t value for 30 degrees of freedom at a significance level of 0.1?

1.3104.

What is the critical t value for 25 degrees of freedom at a significance level of 0.005?

2.7874.

What is the critical t value for 15 degrees of freedom at a significance level of 0.01?

2.6025.

What is the critical t value for 8 degrees of freedom at a significance level of 0.1?

1.3968.

What is the critical F value for 3 numerator and 4 denominator degrees of freedom at a significance level of 0.05?

6.39.

What is the critical F value for 6 numerator and 6 denominator degrees of freedom at a significance level of 0.01?

4.28.

What is the critical t value for 18 degrees of freedom at a significance level of 0.025?

2.1009.

What is the critical t value for 40 degrees of freedom at a significance level of 0.005?

2.7045.

What is the critical t value for 22 degrees of freedom at a significance level of 0.01?

2.5083.

What is the critical F value for 2 numerator and 3 denominator degrees of freedom at a significance level of 0.1?

9.55.