Probability and Statistics: Linear Regression and Correlation

Using the data in the table, calculate the correlation coefficient $R$ to measure the strength of the relationship between the average length of schooling and life expectancy of the population.

Perform a linear regression on the data and analyze the type of correlation based on the value of the correlation coefficient $R$.

Calculate the correlation coefficient between the two variables given the data (x: 1, 2, 3, 4, 5, 6; y: 2, 4, 7, 9, 12, 14).

Calculate by hand the correlation coefficient $r$ for a given set of bivariate data where the $X$ values are $1, 2, 2, 3$ and the $Y$ values are $1, 2, 3, 6$.

Calculate the correlation for the weights measured in kilograms and grams, noting that the correlation score remains the same despite the change in scale.

The scatter plot shows the relationship between the number of slices of pizza eaten by each member of a football team and the number of laps around the block the player could run immediately after. The equation of the regression line is shown in the graph: $\hat{y} = 10 - 0.67x$. Interpret the slope and y-intercept.

A scatter plot shows the relationship between the amount of sugar added to water and the freshness of flowers. If the regression line is given by the equation $\hat{y} = 180.8 + 15.8x$, interpret the slope and y-intercept.

Interpret the slope and intercept of the least squared regression line given by the equation $\hat{\text{score}} = 52.3 + 10.5 \times \text{hours}$, where 'hours' represents the number of hours students studied for a test.

Using the least squares method, find the equation of the line that best fits the given set of data points (x, y): (1, 1.5), (2, 3.8), (3, 6.7), (4, 9.0), (5, 11.2), (6, 13.6), (7, 16). Calculate the slope and the y-intercept of the line of best fit.

Using Excel, calculate the slope and y-intercept of the line of best fit for the given data points.

Calculate the least squares regression equation $\hat{y} = \beta_0 + \beta_1 x$ for the given small data set by hand. Find the estimates for the intercept $\beta_0$ and the slope $\beta_1$.

Given a set of data where the X values represent the number of questions correct out of a possible 20, and the Y values represent students' attitude percentages towards taking tests, use Simple Linear Regression to predict a student's attitude (Y) given a score (X). Specifically, calculate the slope (b) of the regression line using Pearson's correlation coefficient and standard deviations, determine the Y intercept (a), and use these to form the regression equation $y = a + bx$.

Given a training data set containing values for independent variable $x$ (e.g., height of a person) and dependent variable $y$ (e.g., weight), use linear regression to find the linear function $g(x) = \alpha x + \beta$ that best predicts $y$ from $x$.

This involves finding the values of $\alpha$ (slope) and $\beta$ (intercept) that minimize the sum of squared differences between the observed values and the values predicted by the function.

Using residual plots, determine if the normality, constant variance, and linearity assumptions of a simple linear regression model are upheld for a given dataset.

Evaluate the residual plot for the activation level in the brain's pain centres versus their score on the empathic concern scale for 16 women, and determine if the assumptions of the regression model are met.

Evaluate the residual plot for the Janka hardness versus density data for 36 Australian trees, and determine if the assumptions of the regression model are met.

Calculate the residual for a point where the explanatory variable $x$ is represented by hours and the response variable $y$ is represented by the number of people attending a lecture. If the actual number of people at 3 hours is 50 and the predicted number of people at 3 hours using the best fit line is 45.46, what is the residual?