Question 1

A manufacturing company is interested in predicting the number of defects that will be produced each hour on the assembly line.The managers believe that there is a relationship between the defect rate and the production rate per hour.The managers believe that they can use production rate to predict the number of defects.The following data were collected for 10 randomly selected hours.
  Given these sample data,the simple linear regression model for predicting the number of defects is approximately  = 5.67 + 0.048x.

Accepted Answer

The statement describes a situation where a simple linear regression model can be used to predict the number of defects using the production rate as a predictor variable. The given equation represents the estimated linear relationship between the number of defects (the response variable) and the production rate (the predictor variable).

Question 2

When the intercept in a regression equation is deemed not significantly different from 0,then in making predictions for y,0.0 should be used as the value of the intercept rather than the estimated intercept value.

Accepted Answer

The intercept being not significantly different from 0 does not imply it should be set to 0.0 for predictions. The estimated intercept, even if small or statistically insignificant, is part of the model derived from the data and should be used as is for predictions.

Question 3

In analyzing the relationship between two variables,a scatter plot can be used to detect which of the following?

Accepted Answer

A scatter plot can be used to detect all three types of relationships between two variables: positive linear, negative linear, and curvilinear. Therefore, the best choice is D, which includes all of the above options.

Question 4

If the R-squared value for a regression model is high,the regression model will necessarily provide accurate forecasts of the y variable.

Accepted Answer

While a high R-squared value indicates that there is a strong relationship between the independent variables and the dependent variable, it does not necessarily guarantee accurate forecasts. Other factors such as outliers, multicollinearity, or omitted variables may affect the accuracy of the forecasts. It is always important to assess the model's performance using other metrics, such as mean squared error or root mean squared error.

Question 5

A study was recently performed by the Internal Revenue Service to determine how much tip income waiters and waitresses should make based on the size of the bill at each table.A random sample of bills and resulting tips were collected and the following regression results were observed:
SUMMARY OUTPUT  Given this output,the upper limit for the 95 percent confidence interval estimate for the true regression slope coefficient is approximately 0.28.

Accepted Answer

The upper limit for the 95% confidence interval estimate for the true regression slope coefficient can be found in the column "95% Conf. Interval for B" and is equal to 0.2801. Therefore, the statement is true.

Question 6

State University recently randomly sampled seven students and analyzed grade point average (GPA)and number of hours worked off-campus per week.The following data were observed:
  In testing the significance of the regression slope coefficient for the independent variable,HOURS,the calculated test statistic is approximately t = -1.47.

Accepted Answer

Since a test statistic is provided, it indicates that a hypothesis test has been conducted. The null hypothesis is generally stated as the regression slope coefficient being equal to zero, and the alternative hypothesis is that it is not equal to zero. Since the calculated test statistic is not provided with a p-value, we cannot make a statement about the level of significance, but we can state that the test statistic provides evidence against the null hypothesis if it falls in the rejection region. Given that the calculated test statistic is -1.47, we would reject the null hypothesis if the critical t-value at the desired level of significance is less than -1.47. Therefore, Answer A is correct.

Question 7

If a simple least squares regression model is developed based on a sample where the two variables are known to be positively correlated,the sign on the regression coefficient will be positive also.

Accepted Answer

This statement is true. In a simple linear regression model, the sign of the regression coefficient will be positive if the correlation between the two variables is positive, indicating that as one variable increases, so does the other. Conversely, if the correlation is negative, the sign on the regression coefficient will be negative, indicating that as one variable increases, the other decreases.

Question 8

When regression analysis is used for descriptive purposes,two of the main items of interest are whether the sign on the regression slope coefficient is positive or negative and whether the regression slope coefficient is significantly different from zero.

Accepted Answer

This statement is true. When regression analysis is used for descriptive purposes, the sign on the regression slope coefficient indicates the direction of the relationship between the variables, and whether the slope coefficient is significantly different from zero indicates the strength and reliability of the relationship.

Question 9

In a simple linear regression analysis,if the test statistic for testing the significance of the regression slope coefficient is 3.6,the F ratio from the analysis of variance table is known to be 12.96

Accepted Answer

The F ratio from the analysis of variance table is related to the test statistic for the significance of the regression slope coefficient by the following formula:
F = (test statistic)^2 = 3.6^2 = 12.96

Question 10

A study was recently performed by the Internal Revenue Service to determine how much tip income waiters and waitresses should make based on the size of the bill at each table.A random sample of bills and resulting tips were collected and the following regression results were observed:
SUMMARY OUTPUT  Given this output,the point estimate for the average tip per dollar amount of the bill is approximately $0.21.

Accepted Answer

The coefficient for the Bill variable in the regression is 0.21, which represents the estimated average increase in tip amount for every additional dollar on the bill. Therefore, the point estimate for the average tip per dollar amount of the bill is indeed approximately $0.21.

Question 11

The values of the regression coefficients are found such the sum of the residuals is minimized.

Accepted Answer

The values of the regression coefficients are found such that the sum of the squares of the residuals is minimized, not the sum of the residuals themselves. This method is known as the least squares method.

Question 12

If a simple least squares regression model is developed based on a sample where the two variables are known to be positively correlated,the sum of the residuals will be positive.

Accepted Answer

If the two variables are known to be positively correlated, then the least squares regression line will have a positive slope. This means that some residuals will be positive and some will be negative, and their sum will be close to zero if the regression line is a good fit for the data. Therefore, the sum of the residuals can be positive, negative or close to zero, depending on the data and the regression line.

Question 13

The prediction interval developed from a simple linear regression model will be at its narrowest point when the value of x used to predict y is equal to the mean value of x.

Accepted Answer

When the value of x used to predict y is equal to the mean value of x, there is the least amount of deviation from the line of best fit, resulting in the narrowest prediction interval.

Question 14

When calculating prediction intervals for predicted values of y based on a given x,all 95 percent prediction intervals will be of equal width.

Accepted Answer

The width of prediction intervals will depend on the variability of the data and the distance of the predicted value from the mean. Therefore, prediction intervals will not always have the same width.

Question 15

A high coefficient of determination (R²)implies that the regression model will be a good predictor for future values of the dependent variable given the value of the independent variable.

Accepted Answer

A high R² indicates a good fit to the historical data, but it does not guarantee accurate future predictions, as it doesn't account for overfitting or changes in underlying relationships.

Question 16

If the R-square value for a simple linear regression model is .80,the correlation between the two variables is known to be .64.

Accepted Answer

The R-square value is the square of the correlation coefficient (r). If R-square is 0.80, then the correlation (r) would be the square root of 0.80, which is approximately 0.89, not 0.64.

Question 17

A positive population slope of 12 (β₁ = 12)means that a 1-unit increase in x causes an average 12-unit increase in y.

Accepted Answer

The answer of A positive population slope of 12 (β₁...

Question 18

Given the following regression equation,the predicted value for y when x = 0.5 is about 4.57

Accepted Answer

The answer of Given the following regression equation,the predicted value...

Question 19

In simple linear regression,the t-test for the slope and the F-test are both conducting the same hypothesis test.

Accepted Answer

Both tests are used to test the null hypothesis that the slope of the regression line is zero. The t-test is used to test the significance of the slope coefficient, while the F-test is used to test the overall significance of the regression model. However, both tests ultimately test the same hypothesis.

Question 20

A regression model that is deemed to have a regression slope coefficient that could be equal to zero should not be used for prediction since there is no established linear relationship between the x and y variable.

Accepted Answer

A regression model with a slope coefficient that could be equal to zero indicates that there may not be a significant linear relationship between the x and y variables. Therefore, using this model for prediction may not yield accurate results.

Quiz 14: Introduction to Linear Regression and Correlation Analysis