Exam 15: Simple Linear Regression and Correlation

Used cars The following table shows the selling prices and mileages for 7 used cars of a certain model. Mileane Price 22,600 \ 15,200 32,100 \ 12,500 43,800 \ 13,900 51,800 \ 9500 59,400 \ 11,600 65,500 \ 11,700 74,700 \ 8900 -Using the least-squares regression line,predict the selling price of a car with 55,000 miles.

Unemployed The data below are for the number of unemployed persons (in millions)and the federal unemployment insurance payments (in billions of dollars)for the years 1978 - 1985.Some economists state that these two variables are positively related. Year Federal Unemployment Insurance Payments 11.8 10.7 18.0 19.7 23.7 31.5 18.4 16.8 Number of Unemployed Persons 6.2 6.1 7.6 8.3 10.7 10.7 8.5 8.3 -A 95% confidence interval for the slope of the population regression line is (2.168,4.897)Explain how to use the confidence interval to test the hypothesis that the population slope is zero using a 0.05 level of significance.

The residual is defined as the difference between the:

Data values Consider the following data values of x and y. 97 113 81 68 90 79 y 103 103 105 115 127 104 -Use Excel or Minitab to construct a scatter diagram of the data points.

If the coefficient of correlation is -0.90,what percentage of the variation in y is explained by the regression line?

Which of the following statements is true regarding the simple linear regression model $y _ { i } = \beta _ { 0 } + \beta _ { 1 } x _ { i } + \epsilon _ { 1 } ?$

Differentiate between the coefficients of correlation and determination.What information does each one offer that the other does not?

The least-squares criterion requires that the sum of the squared deviations between the y values in the scatter diagram and the y values predicted by the equation be minimized.

Correlation analysis is used to determine the:

A firm finds the coefficient of correlation between y = annual sales and x = annual expenditure on research and development to r = +.095.Comment on the likely direction of causation,if any,between these variables.

Maintenance Costs Data was collected to investigate the relationship between the age and annual maintenance costs for a one particular car.The table below shows the costs over the life of a seven-year old car. Year Annual Maintenance Casts 1 \ 150 2 \ 210 3 \ 450 4 \ 490 5 \ 510 6 \ 1590 7 \ 160 -In testing the hypotheses $H _ { 0 } : \beta _ { 1 } = 0$ vs. $\mathrm { H } _ { 1 } : \beta _ { 1 } \neq 0$ ,what is the rejection region at the 0.02 significance level? What is the conclusion?

Unemployed The data below are for the number of unemployed persons (in millions)and the federal unemployment insurance payments (in billions of dollars)for the years 1978 - 1985.Some economists state that these two variables are positively related. Year Federal Unemployment Insurance Payments 11.8 10.7 18.0 19.7 23.7 31.5 18.4 16.8 Number of Unemployed Persons 6.2 6.1 7.6 8.3 10.7 10.7 8.5 8.3 -What does a scatter diagram of these data indicate about the relationship between these two variables?

Correlation analysis requires that the scales of measurement be expressed in either:

X and Y Consider the following data values of variables x and y x 5 4 3 6 9 8 10 y 7 8 10 5 2 3 1 -Given that when a simple linear regression model applied to the data,it produced the following residuals: 0.062,-0.195,0.548,-0.682,0.089,-0.168,and 0.346. Apply the Lilliefors test to the residuals at $\alpha$ = 0.05.What is your conclusion?

Used cars The following table shows the selling prices and mileages for 7 used cars of a certain model. Mileane Price 22,600 \ 15,200 32,100 \ 12,500 43,800 \ 13,900 51,800 \ 9500 59,400 \ 11,600 65,500 \ 11,700 74,700 \ 8900 -Find the least-squares regression line.

Salesperson The general manager of a chain of furniture 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. Ealesperson Years of Experience Eales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 0 5 14 7 12 20 8 7 17 9 20 30 10 15 25 -Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between years of experience and sales.

An ANOVA test based on SST and SSR is equivalent to t-test for the coefficient of correlation and the slope.

X and Y Consider the following data values of variables x and y x 5 4 3 6 9 8 10 y 7 8 10 5 2 3 1 -Given that the simple linear regression equation is $\hat { y } = 13.223 - 1.257$ ,use Minitab to create a normal probability plot of the residuals.

The simple linear regression model assumes that the y values are statistically independent of each other but the residuals are statistically dependent of each other.

Data values Consider the following data values of x and y. 97 113 81 68 90 79 y 103 103 105 115 127 104 -Create a normal probability plot of the residuals