Exam 16: Simple Linear Regression and Correlat

In the first order linear regression model, the population parameters of the y -intercept and the slope are estimated, respectively, by:

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. {Cost of Books Narrative} Can we infer at the 5% significance level that the editor is correct?

Game Show Winnings & Education An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief she gathers data about the last eight winners of her favorite game show. She records their winnings in dollars and the number of years of education. The results are as follows. {Game Show Winnings & Education Narrative} Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a negative linear relationship exists between years of education and TV game shows' winnings.

A medical statistician wanted to examine the relationship between the amount of sunshine ( x )and incidence of skin discolorations ( y ). As an experiment he found the number of skin discolorations detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. Predict with 95% confidence the skin discolorations per 100,000 in a county with a daily average of 6.5 hours of sunshine.

Suppose the slope of a simple linear regression line between hours studying and exam score is 5. That means as ____________________ increases by one, ____________________ increases by 5.

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: {Oil Quality and Price Narrative} Plot the least squares regression line on the scatter diagram.

Game Show Winnings & Education An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief she gathers data about the last eight winners of her favorite game show. She records their winnings in dollars and the number of years of education. The results are as follows. {Game Show Winnings & Education Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a negative linear relationship exists between years of education and TV game shows' winnings.

Movie Revenues A financier whose specialty is investing in movie productions has observed that, in general, movies with "big-name" stars seem to generate more revenue than those movies 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 movie for ten recently released movies. {Movie Revenues Narrative} Estimate with 95% confidence the average gross revenue of a movie whose top two stars earn $5.0 million.

The standard error of estimate s _e is given by:

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

Consider the following data values of variables x and y . a. Calculate the coefficient of determination, and describe what this statistic tells you about the relationship between the two variables. b. Calculate the Pearson coefficient of correlation. What sign does it have? Why? c. What does the coefficient of correlation calculated tell you about the direction and strength of the relationship between the two variables?

A confidence interval estimate for the expected value of y will always be wider than the prediction interval for the same given value of x and the same confidence level.

{Trivia Games & Education Narrative} Estimate the game winnings for a contestant with 15 years of education.

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. {Theatre Revenues Narrative} Interpret the value of the slope of the regression line.

Game Show Winnings & Education An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief she gathers data about the last eight winners of her favorite game show. She records their winnings in dollars and the number of years of education. The results are as follows. {Game Show Winnings & Education Narrative} Do the tests r and b ₁ in the previous two questions provide the same results? Explain.

The farther a given value of x is from the mean of x , the ____________________ the estimated error becomes.

A store manager gives a pre-employment examination to new employees. The test is scored from 1 to 100. He has data on their sales at the end of one year measured in dollars. He wants to know if there is any linear relationship between pre-employment examination score and sales. An appropriate test to use is the t -test of the population correlation coefficient.

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

Allman Brothers Concert At a recent Allman Brothers 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: {Allman Brothers Concert Narrative} Plot the least squares regression line on the scatter diagram.

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