Exam 16: Simple Linear Regression and Correlation

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?

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. ​ ​ -{Sunshine and Melanoma Narrative} Draw a scatter diagram of the data and plot the least squares regression line on it.

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. ​ ​ -{Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph.

The degrees of freedom for the test statistic for the slope is ____________________.

The graph of a confidence interval for the expected value of y is represented by two parallel lines,one on either side of the regression line.

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} 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?

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. -{Sales and Experience Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a positive linear relationship exists between years of experience and sales.

The symbol for the population coefficient of correlation is:

The standard error of estimate sε is a measure of the:

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

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} Determine the standard error of estimate and describe what this statistic tells you.

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

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. ​ ​ -{Sales and Experience Narrative} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate.

The vertical spread of the data points about the regression line is measured by the y-intercept.

For a regression analysis to be valid,the value of the error variable associated with any particular value of y is ____________________ of the value of the error variable associated with any other value of y.

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} Determine the least squares regression line.

Marc Anthony Concert At a recent Marc Anthony 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: ​ ​ -{Marc Anthony Concert Narrative} Identify possible outliers.

U V'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} Calculate the Pearson coefficient.What sign does it have? Why?

When the actual values y of a dependent variable and the corresponding predicted values are the same,the standard error of estimate sε will be 0.0.

The following data values are for 10 popular types of cereal. ​ ​ a.Determine the least squares regression line,if Calories is the y variable. b.Find the predicted value of Calories for Sugar = 9. c.What does the value of the slope of the regression line tell you? ​