Question

The following sample of observations was randomly selected. x 2 5 3 6 10 7 4 y 4 6 5 7 8 11 6

a. Draw a scatter diagram.
b. Determine mean values and standard deviations for both variables
c. Compute the correlation coefficient
d. Determine regression equation.
e. Determine the value of ŷ when x is 8.
f. Calculate determination coefficient. Considering it’s value make a conclusion about accuracy of the prediction.

Answer 1

The question involves identifying independent and dependent variables, drawing a scatter plot, calculating the least-squares regression line, finding the correlation and determination coefficients, and predicting values. These steps help determine the relationship between variables, the fit of the model, and the accuracy of its predictions.

Step-by-step explanation:

Scatter Plot and Regression Analysis

When addressing a question involving scatter plots and regression, you typically follow a structured approach:

1. Identify the independent and dependent variables: The variable that is being manipulated or changed is the independent variable, and the variable being measured or tested against is the dependent variable.
2. Draw a scatter plot: A scatter plot helps in visually assessing the relationship between the two variables. Each point on the plot corresponds to one observation of the independent (x-axis) and dependent (y-axis) variables.
3. Compute the least-squares regression line: The equation of the line is usually given as ý = a + bx, where 'a' is the y-intercept and 'b' is the slope, derived from the data.
4. Correlation coefficient: This statistical measure, often denoted as 'r', tells us the strength and direction of a linear relationship between two variables.
5. Significance of the correlation coefficient: If the value of 'r' is close to 1 or -1, it indicates a strong linear relationship. The significance of 'r' typically is tested with a hypothesis test or p-value.
6. Regression equation prediction: Using the regression equation, you can predict the value of the dependent variable for a given value of the independent variable, denoted as ŷ when x is given.
7. Determination coefficient: Also known as R-squared, it provides insight into how well the regression model fits the data. It is the square of the correlation coefficient, and a higher value indicates a better fit.

By following these steps, you are equipped to discuss the relationship between variables, the fit of a regression model, and the accuracy of predictions made using the model.