Exam 16: Simple Linear Regression and Correlation

In testing the hypotheses: H₀: $\beta$ ₁ = 0 vs. H₀: $\beta$ ₁ $\neq$ 0, the following statistics are available: , , , , and . The value of the test statistic is:

The width of the confidence interval estimate for the predicted value of y depends on

If there is no linear relationship between two variables x and y, the coefficient of determination must be -1.0.

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

If the regression line is horizontal, the slope is ____________________ and x and y are not related.

In order to estimate with 95% confidence the expected value of y for a given value of x in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

A regression analysis between sales (in $) and advertising (in $) resulted in the following least squares line: . This implies that an increase of $1 in advertising is associated with an increase of $60 in sales.

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. -Determine the least squares regression line.

The residual r_i is defined as the difference between the actual value y_i and the estimated value .

Car Speed and Gas Mileage: An economist wanted to analyze the relationship between the speed of a car (x) and its gas mileage (y). As an experiment a car is operated at several different speeds and for each speed the gas mileage is measured. These data are shown below. -Calculate the coefficient of determination and interpret its value.

If the regression line is horizontal, then we conclude that y ____________________ (is/is not) related to x.

Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a positive linear relationship exists between the quality of oil and price per barrel.

SSE stands for ____________________ of squares for ____________________.

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0.

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: -Estimate with 95% confidence the average oil price per barrel for an API degree of 35.

The confidence interval estimate of the expected value of y for a given value x, compared to the prediction interval of y for the same given value of x and confidence level, will be:

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. -Can we infer at the 5% significance level that the editor is correct?

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: -Determine the least squares regression line.

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. -Determine the least squares regression line.

Wayne Newton Concert: At a recent Wayne Newton concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year. The following data were collected: An Excel output follows: -Estimate with 95% confidence the average number of concerts attended by all 45 year-old individuals.