Exam 16: Simple Linear Regression and Correlat

Accidents and Rain A statistician investigating the relationship between the amount of rain (in inches)and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year. The results are shown below. {Accidents and Rain Narrative} Find the least squares regression line.

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} Determine the least squares regression line.

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. {UV's and Skin Cancer Narrative} What does the coefficient of correlation calculated tell you about the direction and strength of the relationship between the two variables?

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} Which interval in the previous two questions is narrower: the confidence interval estimate of the expected value of y or the prediction interval for the same given value of x (10 years)and same confidence level? Why?

Income and Education A professor of economics wants to study the relationship between income ( y in $1000s)and education ( x in years). A random sample eight individuals is taken and the results are shown below. {Income and Education Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.

A regression analysis between sales (in $1000)and advertising (in $100)resulted in the following least squares line: . This implies that if advertising is $800, then the predicted amount of sales (in dollars)is:

In a simple linear regression model b ₁ is the ____________________ of the straight line.

In simple linear regression, most often we perform a two-tail test of the population slope b ₁ to determine whether there is sufficient evidence to infer that a linear relationship exists. The null hypothesis is stated as:

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:

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} Predict with 95% confidence the gross revenue of a movie whose top two stars earn $5.0 million.

Game 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 Winnings & Education Narrative} Estimate with 95% confidence the average winnings of all contestants who have 10 years of education.

In simple linear regression, the denominator of the standard error of estimate s _e is .

Truck Speed vs 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. {Truck Speed vs Gas Mileage Narrative} Determine the least squares regression line.

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

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

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. {Rock Concert  Revenues Narrative} Calculate the Pearson correlation coefficient. Interpret the results.

A scatter diagram includes the following data points: Two regression models are proposed: (1) , and (2) . Using the least squares method, which of these regression models provides the better fit to the data? Why?

The method statisticians use to produce a straight line that minimizes the SSE is called the ____________________ method.

In order to predict with 90% 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?

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