Question 1

In general,increasing the variation of x will lead to a higher coefficient of determination.

Accepted Answer

Increasing the variation of x will lead to a wider range of data points, which can help to better fit a regression line to the data and increase the overall goodness of fit as measured by the coefficient of determination.

Question 2

The coefficient of determination is the proportion of the total variation in the independent variable X that is explained or accounted for by its relationship with the dependent variable Y.

Accepted Answer

The coefficient of determination is the proportion of the total variation in the dependent variable Y that is explained or accounted for by its relationship with the independent variable X.

Question 3

The coefficient of determination can only take on positive values.

Accepted Answer

The coefficient of determination, also known as R-squared, is a measure of how well an observed value can be predicted by a model. It ranges from 0 to 1, indicating the proportion of the variation in the dependent variable that is explained by the independent variable(s). Since it is a squared quantity, it can only be positive or zero.

Question 4

When testing the strength of the relationship between two variables,the null hypothesis of interest is H₀ : ρ = 0.

Accepted Answer

The null hypothesis for testing the strength of the relationship between two variables is usually stated as H0: ρ = 0, where ρ represents the population correlation coefficient. Therefore, the statement in the question is true.

Question 5

In the sample regression line = b₀ + b₁x,the term b₀ is the y-intercept;this is the value of y where the line intersects the y-axis whenever x = 0.

Accepted Answer

The statement correctly explains that the term b₀ is the y-intercept, which is the value of y when x = 0. Therefore, the statement is true.

Question 6

A correlation coefficient equal to -1 or +1 indicates perfect correlation.

Accepted Answer

A correlation coefficient of -1 or +1 represents perfect negative or positive correlation, respectively. This means that the two variables are perfectly linearly related, with no variability in the data points.

Question 7

You give a pre-employment examination to your applicants.The test is scored from 1 to 100.You have data on their sales at the end of one year measured in dollars.You want to know if there is any linear relationship between pre-employment examination score and sales.An appropriate test to use is the t-test on the population correlation coefficient.

Accepted Answer

The t test on the population correlation coefficient is used to test the hypothesis that there is or is not a significant linear relationship between two variables. In this case, the variables are the pre-employment examination score and sales, and the test is appropriate to determine if there is a linear relationship between these variables or not.

Question 8

The coefficient of determination is a statistical test of the fit of the linear regression.

Accepted Answer

The answer of The coefficient of determination is a statistical...

Question 9

The smaller the sample size,the smaller the standard error of estimate.

Accepted Answer

This statement is false. The standard error of estimate is actually inversely proportional to the sample size. As the sample size increases, the standard error of estimate decreases, and vice versa.

Question 10

The width of the confidence interval estimate for the average value of y does not depend on the standard error of the estimate.

Accepted Answer

The width of the confidence interval estimate for the average value of y is directly proportional to the standard error of the estimate. The larger the standard error, the wider the interval.

Question 11

If the correlation coefficient is greater than 0.5 then the coefficient of determination R² from a simple regression model is greater than 0.25.

Accepted Answer

This is true because the coefficient of determination (also known as R-squared) is equal to the square of the correlation coefficient, meaning that if the correlation is greater than 0.5, the coefficient of determination will be greater than 0.25 (0.5 squared is 0.25).

Question 12

The coefficient of determination is often interpreted as the percent of variability in the dependent variable y that is explained by the regression equation.

Accepted Answer

The coefficient of determination, denoted by R², represents the proportion of the variance in the dependent variable that is explained by the independent variable(s) in the regression equation. This value can range from 0 to 1, or 0% to 100%. Therefore, it is often interpreted as the percentage of variability in the dependent variable y that is explained by the regression equation.

Question 13

The divisor of the standard error of estimate in a simple linear regression is n - 2.

Accepted Answer

The divisor for the standard error of estimate in a simple linear regression is actually n - 1, not n - 2. This is because one degree of freedom is used for the slope coefficient and one for the intercept coefficient, leaving n - 2 degrees of freedom for the error term. However, the correct statement that the divisor is n - 1 still leads to the conclusion that choice A is true.

Question 14

The value of the coefficient of determination can never be larger than 1.0.

Accepted Answer

The coefficient of determination, also known as R-squared, is a statistical measure that ranges between 0 and 1, where 1 indicates a perfect fit of the data to the regression line or model. Therefore, the value of the coefficient of determination can never be larger than 1.0.

Question 15

The coefficient of determination is the positive square root of the coefficient of correlation.

Accepted Answer

The coefficient of determination is actually the squared value of the coefficient of correlation. Therefore, the correct statement is that the coefficient of correlation is the positive or negative square root of the coefficient of determination.

Question 16

The coefficient of determination represents the ratio of SSR to SST.

Accepted Answer

The coefficient of determination, also known as R-squared, is equal to SSR/SST, where SSR is the sum of squares due to regression and SST is the total sum of squares. Therefore, the statement is true.

Question 17

When testing the strength of the relationship between two variables,the alternate hypothesis is H₁ ;ρ ≠ 0.

Accepted Answer

The alternate hypothesis for testing the strength of the relationship between two variables states that the population correlation coefficient (ρ) is not equal to 0. This is reflected in the statement H₁ ;ρ ≠ 0, where the semicolon represents "given that" and the ellipsis before and after the statement indicate that additional context may be present. Therefore, the answer is A)True.

Question 18

Suppose that for two random variables,x and y,we test H₀ : ρ = 0 against a two-sided alternative,and we are unable to reject the null hypothesis.We could therefore conclude that there is no relationship between x and y.

Accepted Answer

Failing to reject the null hypothesis does not necessarily mean that there is no relationship between x and y. It only means that there is not enough evidence to reject the null hypothesis at the given significance level. There could still be a relationship between x and y that is not detected in the current sample.

Question 19

In simple linear regression,the fit of the regression equation to the data is improved as error sum of squares increases and regression sum of squares decreases.

Accepted Answer

In simple linear regression, the fit of the regression equation to the data is improved as regression sum of squares increases and error sum of squares decreases.

Question 20

When we compute the sample correlation r from data,the result will be definitely zero when the population correlation ρ is zero.

Accepted Answer

It is possible for the sample correlation r to be zero even when the population correlation ρ is not zero. This can happen when there is random variation in the sample that masks the true correlation in the population.

Quiz 11: Simple Regression