Statistics and R Programming Basics

The mean function calculates the average of the numbers in the vector.

You can specify the base by using named arguments, such as log(x=4, base=2).

The c function creates a vector, which is a sequence of values of the same type.

<p>It is represented as $X ∼ N(µ, σ^2)$ where $µ$ is the mean and $σ^2$ is the variance.</p> Signup and view all the answers

<p>You use the command <code>plot(x=c(2,3), y=c(3,4))</code> to display the points.</p> Signup and view all the answers

<p>A Poisson distribution models the number of events in a fixed interval, with mean $E(Y) = λ$.</p> Signup and view all the answers

<p><code>sqrt</code> represents the square root function in R and is used as <code>sqrt(x)</code>.</p> Signup and view all the answers

<p>A discrete random variable takes on a finite number of possible values, each with a specific probability.</p> Signup and view all the answers

<p>The likelihood function is the probability or density function of the observed data x, viewed as a function of the unknown parameter θ.</p> Signup and view all the answers

<p>The maximum likelihood estimate (MLE) ˆθML is the point where the likelihood function assumes its maximum value.</p> Signup and view all the answers

<p>A confidence interval is a range of estimates for an unknown parameter that represents the long-run proportion of intervals containing the true value at a designated confidence level.</p> Signup and view all the answers

<p>The main purpose of classification is to predict qualitative responses by assigning observations to categories or classes.</p> Signup and view all the answers

<p>Fixed variables, known as explanatory or covariates, are the known inputs, while the random variable, referred to as the response, is the output being predicted.</p> Signup and view all the answers

<p>Logistic regression predicts the probability that an observation belongs to a particular category, especially for binary qualitative responses.</p> Signup and view all the answers

<p>Simple linear regression predicts a quantitative response Y based on a single predictor variable X, assuming a linear relationship between them.</p> Signup and view all the answers

<p>The notation ‘≈’ indicates that the response variable Y is approximately modeled as a linear function of the predictor variable X.</p> Signup and view all the answers

<p>Yes, classification problems often occur more frequently than regression problems in many real-world applications.</p> Signup and view all the answers

<p>The sample space T is the set of all possible realizations of the random variable X.</p> Signup and view all the answers

<p>Training observations provide the necessary data to create a classifier by allowing it to learn the relationship between input features and output categories.</p> Signup and view all the answers

<p>Maximum likelihood estimation identifies plausible values of θ by favoring those that give relatively high likelihood to observed data.</p> Signup and view all the answers

<p>In healthcare, classification can be applied to determine which medical condition a patient has based on their symptoms.</p> Signup and view all the answers

<p>Encoding qualitative responses can lead to fundamentally different linear models, resulting in diverse predictions on test observations.</p> Signup and view all the answers

<p>Online banking uses classification methods to determine whether a transaction is fraudulent based on user data like IP address and transaction history.</p> Signup and view all the answers

<p>Identifying deleterious DNA mutations helps to distinguish between disease-causing mutations and non-harmful ones, impacting patient treatment plans.</p> Signup and view all the answers

<p>As the number of observations (n) increases, the standard error of the estimate decreases.</p> Signup and view all the answers

<p>The residual standard error (RSE) is an estimate of the standard deviation of the errors or residuals in a regression model, and it is used to assess the accuracy of the model's predictions.</p> Signup and view all the answers

<p>A 95% confidence interval is a range of values that is expected to contain the true parameter value with 95% probability, providing a measure of uncertainty in the estimates.</p> Signup and view all the answers

<p>ESS measures the variability explained by the regression model, while RSS measures the variability of the residuals. Their relationship is defined by the equation TSS = ESS + RSS.</p> Signup and view all the answers

<p>A higher coefficient of determination ($R^2$) signifies that a greater proportion of the variability in the dependent variable is explained by the regression model.</p> Signup and view all the answers

<p>Assessing the goodness of fit is important to determine how well the model predictions align with observed data and to evaluate the model's effectiveness.</p> Signup and view all the answers

<p>The variance of the residuals indicates the spread of the errors; low variance suggests that the model's predictions are close to the observed values.</p> Signup and view all the answers

<p>Standard errors can be used to compute confidence intervals for parameter estimates and to conduct hypothesis tests about the significance of those parameters.</p> Signup and view all the answers

<p>A small p-value indicates that it is unlikely to observe the association between X and Y due to chance, suggesting a potential relationship.</p> Signup and view all the answers

<p>The typical cutoff values for rejecting the null hypothesis are 5% or 1%.</p> Signup and view all the answers

<p>The assumption of causality implies that a causal relationship exists, allowing for one variable to be considered a response to another explanatory variable.</p> Signup and view all the answers

<p>High variability in residuals indicates that the data points are far from the regression line, suggesting a poor fit.</p> Signup and view all the answers

<p>The steps include model specification and assumptions, point estimation, interpretation of coefficients, calculation of standard errors, and diagnostics.</p> Signup and view all the answers

<p>Multiple linear regression accommodates several explanatory variables, whereas simple linear regression involves only one predictor.</p> Signup and view all the answers

<p>Correlation measures the strength and direction of the relationship, indicating how reliable the association is between the two variables.</p> Signup and view all the answers

<p>It is important to check assumptions to ensure the validity of the model and the reliability of the regression results.</p> Signup and view all the answers

<p>The purpose of the least square estimate is to find the value of β that minimizes the sum of squared discrepancies between the observed values and the fitted values.</p> Signup and view all the answers

<p>The residual sum of squares is the minimum value of the squared discrepancies between the observed values and the fitted values, indicating the model's accuracy.</p> Signup and view all the answers

<p>The fitted value vector yˆ represents the linear combination of the columns of X that minimizes the squared distance from the observed data y.</p> Signup and view all the answers

<p>The model for two-group comparisons can be represented as y = Xβ + ϵ, where y is the response variable, X is the design matrix, and β represents the group means.</p> Signup and view all the answers

<p>Computing F-statistics helps to determine whether at least one of the predictors is useful in predicting the response variable.</p> Signup and view all the answers

<p>In simple linear regression, we can check for a relationship by determining if β1 is not equal to zero.</p> Signup and view all the answers

<p>A key question to consider is whether all the regression coefficients are zero, i.e., whether β1 = β2 = ... = βp = 0.</p> Signup and view all the answers

<p>A low p-value indicates that at least one predictor variable is significantly associated with the response variable.</p> Signup and view all the answers