Exam 16: Simple Linear Regression and Correlation

If the coefficient of correlation is 0.90, then the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is:

Determine the coefficient of determination and discuss what its value tells you about the two variables.

When all the actual values of y are equal to their predicted values, the standard error of estimate will be:

Rock Concert Revenues: A financier whose specialty is investing in rock concerts has observed that, in general, concerts with "big-name" stars seem to generate more revenue than those concerts whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid performers in the concert for ten concert tours. -Calculate the Pearson correlation coefficient. Interpret the results.

In a simple linear regression problem, the following sum of squares are produced: , , and . The percentage of the variation in y that is explained by the variation in x is:

Cost of Books: The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyze the claim, a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected books. These data are listed below. -Predict with 90% confidence the selling price of a book with 900 pages.

If you take a particular x value and plug it into a regression line equation, the result is a(n) ____________________ estimate for y.

Sunshine and Melanoma: A medical researcher wanted to examine the relationship between the amount of sunshine (x) in hours, and incidence of melanoma, a type of skin cancer (y). As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. -Determine the least squares regression line.

The least squares method for determining the best fit minimizes:

The smallest value that the standard error of estimate s _{$\varepsilon$} can assume is:

Cost of Textbooks: The editor of a higher education book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger textbooks will cost more money. As an experiment to analyze the claim, a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected textbooks. These data are listed below. -Interpret the value of the slope of the regression line.

In the least squares regression line , the predicted value of y equals:

There is more error in estimating a mean value of y as opposed to predicting an individual value of y.

Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y.

Sales and Experience: The general manager of a chain of department stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. -Determine the standard error of estimate and describe what this statistic tells you about the regression line.

A regression analysis between weight (y in pounds) and height (x in inches) resulted in the following least squares line: . This implies that if the height is increased by 1 inch, the weight, on average, is expected to:

UV's and Skin Cancer: A medical statistician wanted to examine the relationship between the amount of UV's (x) and incidence of skin cancer (y). As an experiment he found the number of skin cancers detected per 100,000 of population and the average daily sunshine in eight states around the country. These data are shown below. -Calculate the Pearson coefficient. What sign does it have? Why?

Theatre Revenues: A financier whose specialty is investing in stage productions has observed that, in general, movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid performers in the play for ten recently staged plays. -Estimate the gross revenue of a play if the two highest paid performers received 6 million dollars.

The point where confidence intervals and prediction intervals do best is .

Oil Quality and Price: Quality of oil is measured in API gravity degrees--the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel. A partial Minitab output follows: -Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate to describe the relationship between the quality of oil and price per barrel.