Question 1

The fitted intercept in a regression has little meaning if no data values near X = 0 have been observed.

Accepted Answer

The fitted intercept represents the value of the dependent variable when the independent variable has a value of 0. Therefore, if no data values near X = 0 have been observed, the fitted intercept does not have a meaningful interpretation.

Question 2

The least squares regression line is obtained when the sum of the squared residuals is minimized.

Accepted Answer

This is the fundamental principle behind least squares regression. The line of best fit is determined when the sum of the squared differences between the predicted values and the observed values is minimized.

Question 3

If R² = .36 in the model Sales = 268 + 7.37 Ads with n = 50,the two-tailed test for correlation at α = .05 would say that there is a significant correlation between Sales and Ads.

Accepted Answer

To test for correlation between Sales and Ads, we need to calculate the correlation coefficient (r) using the formula: r = (nΣXY - ΣXΣY) / sqrt[(nΣX^2 - (ΣX)^2)(nΣY^2 - (ΣY)^2)] where n is the sample size, ΣX is the sum of X values, ΣY is the sum of Y values, ΣXY is the sum of the product of X and Y values, ΣX^2 is the sum of squared X values, and ΣY^2 is the sum of squared Y values. In this case, we are given R² = .36, which is the value of r. We can use this value to calculate the test statistic t, using the formula: t = r * sqrt(n - 2) / sqrt(1 - r^2) Using n = 50 and α = .05 (which corresponds to a critical t-value of ±2.01), we can find the rejection region for the two-tailed test. If the calculated t-value falls outside this region, we reject the null hypothesis of no correlation. Plugging in the values, we get t = 4.39, which is greater than the critical t-value of ±2.01. Therefore, we reject the null hypothesis and conclude that there is a significant correlation between Sales and Ads. Thus, the answer is A.

Question 4

If R² = .36 in the model Sales = 268 + 7.37 Ads,then Ads explains 36 percent of the variation in Sales.

Accepted Answer

The answer of If R² = .36 in the model...

Question 5

If SSR is 1800 and SSE is 200,then R² is .90.

Accepted Answer

SSR (Sum of Squares Regression) = 1800 SSE (Sum of Squares Error) = 200 The formula to calculate R-squared (R²) is: R-squared = SSR / (SSR + SSE) Plugging in the values we have: R-squared = 1800 / (1800 + 200) R-squared = 1800 / 2000 R-squared = 0.9 Therefore, the statement is true and the answer is A.

Question 6

The ordinary least squares regression line always passes through the point  .

Accepted Answer

The answer of The ordinary least squares regression line always...

Question 7

In the model Sales = 268 + 7.37 Ads,an additional $1 spent on ads will increase sales by 7.37 percent.

Accepted Answer

An additional $1 spent on ads will increase sales by 7.37 units, not 7.37 percent.

Question 8

A sample correlation r = .40 indicates a stronger linear relationship than r = -.60.

Accepted Answer

The absolute value of the correlation coefficient (|.40| = .40) indicates a weaker relationship than the absolute value of -.60 (|.60| = .60). However, if we only look at the signs, we can say that the relationship is stronger in the negative direction for r = -.60.

Question 9

A common source of spurious correlation between X and Y is when a third unspecified variable Z affects both X and Y.

Accepted Answer

This is true because when Z affects both X and Y, there may appear to be a correlation between X and Y, but it is actually due to the third variable Z. This is known as a confounding variable.

Question 10

The correlation coefficient r measures the strength of the linear relationship between two variables.

Accepted Answer

The statement is true.

Question 11

A scatter plot is used to visualize the association (or lack of association)between two quantitative variables.

Accepted Answer

This statement is true. A scatter plot shows the relationship between two variables by displaying data points as dots on a two-dimensional plane. Each dot represents a combination of values for both variables, allowing for a visual examination of the possible association between them.

Question 12

In a simple regression,if the coefficient for X is positive and significantly different from zero,then an increase in X is associated with an increase in the mean (i.e. ,the expected value)of Y.

Accepted Answer

A positive coefficient for X indicates a positive relationship between X and Y. A significant difference from zero indicates that this relationship is unlikely to be due to chance. Therefore, an increase in X is associated with an increase in the mean (expected value) of Y in a simple regression with a positive coefficient for X.

Question 13

In least-squares regression,the residuals e₁,e₂,... ,e_n will always have a zero mean.

Accepted Answer

This statement is true. In least-squares regression, the residual is defined as the difference between the observed value and the predicted value of the response variable. Since the equation for the least-squares line is constructed to minimize the sum of the squared residuals, the residuals will always average out to zero over the entire dataset. In other words, any positive residuals will be counterbalanced by negative residuals, resulting in an average of zero. Therefore, the residuals have a zero mean.

Question 14

Pearson's correlation coefficient (r)requires that both variables be interval or ratio data.

Accepted Answer

Pearson's correlation coefficient measures the linear relationship between two variables that are measured on an interval or ratio scale. It cannot be calculated for categorical or nominal data.

Question 15

The correlation coefficient r always has the same sign as b₁ in Y = b₀ + b₁X.

Accepted Answer

The correlation coefficient r has the same sign as the slope coefficient b₁ in Y = b₀ + b₁X.

Question 16

The least squares regression line gives unbiased estimates of β₀ and β₁.

Accepted Answer

The least squares regression line minimizes the sum of squared residuals, which ensures that the estimates of β₀ and β₁ are unbiased.

Question 17

In a simple regression,the correlation coefficient r is the square root of R².

Accepted Answer

The answer of In a simple regression,the correlation coefficient r...

Question 18

The width of a prediction interval for an individual value of Y is less than standard error s_e.

Accepted Answer

The width of a prediction interval for an individual value of Y is typically greater than the standard error s, as it also takes into account the variability of the predictor variable(s). Therefore, the statement is false.

Question 19

If r = .55 and n = 16,then the correlation is significant at α = .05 in a two-tailed test.

Accepted Answer

To determine if a correlation is significant, we need to calculate the t-value using the formula t = r√(n-2) / √(1-r^2). Substituting the given values, we get t = 2.306. With 14 degrees of freedom (n-2), the critical t-value for a two-tailed test at α = .05 is 2.145. Since the calculated t-value is greater than the critical t-value, we can reject the null hypothesis and conclude that the correlation is significant at α = .05.

Question 20

When using the least squares method,the column of residuals always sums to zero.

Accepted Answer

The least squares method involves minimizing the sum of squared residuals, which means that the column of residuals must sum to zero in order to achieve this minimum. This is a property of the least squares method and holds true regardless of the specific problem or data set being analyzed.

Quiz 12: Simple Regression

The fitted intercept in a regression has little meaning if no data values near X = 0 have been observed.

The least squares regression line is obtained when the sum of the squared residuals is minimized.

If R2 = .36 in the model Sales = 268 + 7.37 Ads with n = 50,the two-tailed test for correlation at α = .05 would say that there is a significant correlation between Sales and Ads.

If R2 = .36 in the model Sales = 268 + 7.37 Ads,then Ads explains 36 percent of the variation in Sales.

If SSR is 1800 and SSE is 200,then R2 is .90.

The ordinary least squares regression line always passes through the point .