Exam 10: Correlation and Regression

Describe the standard error of estimate, se. How do smaller values of se relate to the dispersion of data points about the line determined by the linear regression equation? What does it mean when se is 0?

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximation: $r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$ where $t$ is the $t$ -score from the $t$ Distribution table corresponding to $n - 2$ degrees of freedom. Use this approximation to find critical values of $r _ { \mathrm { S } }$ for the case where $n = 40$ and $\alpha = 0.10$ .

Explain what is meant by the coefficient of determination, $\mathrm { r } ^ { 2 }$ . Give an example to support your result.

Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical values of r given the number of pairs of data n and the significance level $\alpha$ - $\mathrm { n } = 19 , \alpha = 0.01$

Use the given data to find the best predicted value of the response variable. -The regression equation relating dexterity scores $( \mathrm { x } )$ and productivity scores $( \mathrm { y } )$ for the employees of a company is $\hat{y} = 5.50 + 1.91 x$ . Ten pairs of data were used to obtain the equation. The same data yield $\mathrm { r } = 0.986$ and $\overline { \mathrm { y } } = 56.3$ . What is the best predicted productivity score for a person whose dexterity score is $37 ?$

Use the rank correlation coefficient to test for a correlation between the two variables. -Given that the rank correlation coefficient, r $\mathrm { r } _ { \mathrm { S } },$ , for 33 pairs of data is 0.338, test the claim of correlation between the two variables. Use a significance level of 0.01.

Use the rank correlation coefficient to test for a correlation between the two variables. -Given that the rank correlation coefficient, rs, for 73 pairs of data is -0.663, test the claim of correlation between the two variables. Use a significance level of 0.05.

Six pairs of data yield $\mathrm { r } = 0.789$ and the regression equation $\hat { y } = 4 x - 2$ . Also, $\bar { y } = 19.0$ . What is the best predicted value of $y$ for $x = 5$ ?

A set of data consists of the number of years that applicants for foreign service jobs have studied German and the grades that they received on a proficiency test. The following regression equation is obtained: $\hat { \mathrm { y } } = 31.6 + 10.9 \mathrm { x }$ where x represents the number of years of study and y represents the grade on the test. What does the slope of the regression line represent in terms of grade on the test?

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximation: $r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$ where $\mathrm { t }$ is the $\mathrm { t }$ -score from the $t$ Distribution table corresponding to $\mathrm { n } - 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 17$ and $\alpha = 0.05$ .

Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical values of r given the number of pairs of data n and the significance level $\alpha$ - $\mathrm { n } = 25 , \alpha = 0.05$

The regression equation for a set of paired data is $\hat { y } = 9 + 2 x$ . The correlation coefficient for the data is 0.87. A new data point, P(12, 53), is added to the set.

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. -Managers rate employees according to job performance and attitude. The results for several randomly selected employees are given below. Performance 59 63 65 69 58 77 76 69 70 64 Attitude 72 67 78 82 75 87 92 83 87 78

Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear correlation. Use a significance level of 0.05. - $r = - 0.853 , n = 5$

Use the rank correlation coefficient to test for a correlation between the two variables. -Given that the rank correlation coefficient, r $\mathrm { r } _ { \mathrm { S } },$ for 20 pairs of data is 0.827, test the claim of correlation between the two variables. Use a significance level of 0.05.

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. - x 2 4 5 6 y 7 11 13 20

Construct a scatterplot for the given data - 0.55 0.33 0.24 0.2 -0.34 0.57 0.41 0.14 0.68 0.78 0.66 0.34 -0.13 0.95 0.8 -0.12

Construct a scatterplot for the given data. - 2 -7 -2 -8 -9 2 -4 8 -4 -1 -3 -7 -2 4 3 2 1 5 -7 -5

Find the value of the linear correlation coefficient r. - 19.3 23.4 10.3 17.7 23.2 6 2 3 3 3

Use the rank correlation coefficient to test for a correlation between the two variables. -A placement test is required for students desiring to take a finite mathematics course at a university. The instructor of the course studies the relationship between students' placement test score and final course score. A random sample of eight students yields the following data. Placement Score Final Course Score 38 63 90 41 95 54 51 32 86 93 74 60 60 61 57 89 Compute the rank correlation coefficient, rs, of the data and test the claim of correlation between placement score and final course score. Use a significance level of 0.05.