Exam 15: Simple Linear Regression and Correlation

____________________ provides a "best-fit" mathematical equation for the values of two variables,x and y.

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 -Determine the least-squares regression line.

If the coefficient of correlation r is positive,then the dependent variable and the independent variable x are said to be ____________________ related.However,if r is negative,then x and y are said to be ____________________ related.

Differentiate between a confidence interval and a prediction interval.

The coefficient of determination can take on values between ____________________ and ____________________,inclusive.

A simple linear regression problem produced the following sum of squares: $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ = 240, $\sum \left( y _ { i } - \hat { y } _ { i } \right) ^ { 2 }$ = 60, $\sum \left( \\\hat { y } _ { i } - \bar { y } \right) ^ { 2 }$ = 180 What percentage of the variation in y is explained by the regression line?

The coefficient of correlation in a sample of 5 is 0.832.Use a 0.10 significance level to test the hypothesis that the population coefficient of correlation is zero.

The value of the variation explained by the regression line can never be larger than 1.0.

The coefficient of determination can be described in terms of the total variation in y versus the unexplained variation in y.

GPA The following table shows the grade point average (GPA)for 5 students along with their entrance exam scores for MBA programs (GMAT).Develop a model that would predict the GPA of a student based on their GMAT score. Student CPA GMAT 1 3.5 6.7 2 3.1 5.4 3 3.2 4.8 4 4.0 7.0 5 3.6 5.9 6 3.2 5.2 7 3.7 7.2 -Determine the least-squares regression line.

In regression and correlation analysis,what are the two components of the total amount of variation in the y values? Define the coefficient of determination in terms of these components.

GPA The following table shows the grade point average (GPA)for 5 students along with their entrance exam scores for MBA programs (GMAT).Develop a model that would predict the GPA of a student based on their GMAT score. Student CPA GMAT 1 3.5 6.7 2 3.1 5.4 3 3.2 4.8 4 4.0 7.0 5 3.6 5.9 6 3.2 5.2 7 3.7 7.2 -A simple linear regression problem produced the following sum of squares: $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ = 240, $\sum \left( y _ { i } - \hat { y } _ { i } \right) ^ { 2 }$ = 60, $\sum \left( \hat { y } _ { i } - \bar { y } \right) ^ { 2 }$ = 180 What percentage of the variation in y is not explained by the regression line?

Teacher A statistics teacher collected the following data to determine if the number of hours a student studied during the semester could be used to predict the final grade for the course. Student Hours Studyine Final Grade 1 42 92 2 58 95 3 32 81 4 39 78 5 37 75 6 51 88 7 49 85 8 45 85 -In testing the hypotheses $H _ { 0 } : \beta _ { 1 } = 0$ vs. $\mathrm { H } _ { 1 } : \beta _ { 1 } \neq 0$ ,what is the rejection region at the 0.05 significance level?

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 -Predict with 95% confidence the monthly sales of a salesperson with 10 years of experience. 95% CI = ________________ Enter as (n1,n2)with no $ or "," in the number.

In a simple linear regression model,the residual is the horizontal distance from the regression line to an observed data point.

At the 0.05 level of significance,test the null hypothesis that the population slope is zero.

Regardless of the value of x,the standard deviation of the distribution of y values about the regression line is the same.This assumption of equal standard deviations about the regression line is called:

The values of y and x are inversely related,and 84.64% of the variation in y is explained by the regression equation.What is the coefficient of correlation?

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 -Determine the coefficient of determination and discuss what its value tells you about the two variables. $\mathrm { r } ^ { 2 }$ = ____________________ Discuss

The value for SSE equals zero.This means that the coefficient of determination (r ² )must equal: