Exam 16: Simple Linear Regression and Correlation

In a regression problem,if the coefficient of determination is 0.95,this means that:

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

The confidence interval estimate of the expected value of y will be narrower than the prediction interval for the same given value of x and confidence level.This is because there is less error in estimating a mean value as opposed to predicting an individual value.

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} Interpret the value of the slope of the regression line.

The standardized residual is defined as:

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} Predict with 95% confidence the monthly sales of a salesperson with 10 years of experience.

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

Comedy Shows Revenues A financier whose specialty is investing in comedy shows has observed that,in general,shows with "big-name" stars seem to generate more revenue than those shows 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 comedians in the show for ten recently staged shows. ​ ​ -{Comedy Shows Revenues Narrative} Use the predicted and actual values of y to calculate the residuals.

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: -{Wayne Newton Concert Narrative} Estimate with 95% confidence the average number of concerts attended by all 45 year-old individuals.

The probability distribution of the error variable ε is normal,with mean E(ε)= 0,and standard deviation σε =1.

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

In performing a regression analysis which of the following must be true about the distribution of the error variable?

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} Determine the standard error of estimate and describe what this statistic tells you about the 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: -{Wayne Newton Concert Narrative} Predict with 95% confidence the number of concerts attended by a 45 year-old individual.

In simple linear regression,most often we perform a two-tail test of the population slope β₁ to determine whether there is sufficient evidence to infer that a linear relationship exists.The null hypothesis is stated as:

Grateful Dead Concert At a recent Grateful Dead 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.It is suspected that older concert goers tend to go to more of his concerts in one year than younger concert goers.The data and analysis are shown below. An Excel output follows: -{Grateful Dead Concert Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a positive linear relationship exists between age and number of concerts attended.

For a regression analysis to be valid,the error variable must have a mean of ____________________.

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

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.

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} What does the coefficient of correlation calculated tell you about the direction and strength of the relationship between the two variables?