Exam 10: Correlation and Regression

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. - $\mathrm { r } = 0.803 , \mathrm { n } = 9$

Below are the productivity, dexterity, and job satisfaction ratings of ten randomly selected employees. Productivity 23 25 28 21 21 25 26 30 34 36 Dexterity 49 53 59 42 47 53 55 63 67 75 Job satisfaction 56 58 60 50 54 61 59 63 67 69 Find the multiple regression equation that expresses the job satisfaction scores in terms of the productivity and d scores.

The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\mathrm { y } = 44.8447 + 3.52427 \mathrm { x }$ . Find the unexplained variation. Hours of preparation 5 2 9 6 10 Test score 64 48 72 73 80

The equation of the regression line for the paired data below is $\hat { y } = 3 x$ . Find the explained variation. 2 4 5 6 7 11 13 20

After obtaining a regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ , a confidence interval for the mean of all values $y$ for which $x =$ $\mathrm { x } _ { 0 }$ can be obtained as follows: $\left( \hat { y } - E _ { r } \hat { y } + E \right) ,$ where $E = \left( t _ { \alpha / 2 } \right) \operatorname { se } \sqrt { \frac { 1 } { n } + \frac { n \left( x _ { 0 } - \bar { x } \right) ^ { 2 } } { n \sum x ^ { 2 } - \left( \sum x \right) ^ { 2 } } }$ The critical value $t _ { \alpha / 2 }$ is found from the $t$ -table using $n - 2$ degrees of freedom. Use the data below to obtain a $95 \%$ confidence interval estimate of the mean test score of all students who study hours. Note that the equation of the regression line is $\hat { y } = 52.8804 + 3.405 \mathrm { x }$ and that $\mathrm { s } _ { \mathrm { e } } = 6.8359$ . (hours studied) 2.5 4.5 5.1 7.9 11.6 (score on test) 66 70 60 83 93

The equation of the regression line for the paired data below is $\hat { y } = 6.1829 + 4.3394 \mathrm { x }$ and the standard error of estimate is $\mathrm { se } _ { \mathrm { e } } = 1.6419$ . Find the $99 \%$ prediction interval of $\mathrm { y }$ for $\mathrm { x } = 12$ . 9 7 2 3 4 22 17 43 35 16 21 23 102 81

Use computer software to obtain the regression and identify R², adjusted R², and the P-value -An anti-smoking group used data in the table to relate the carbon monoxide of various brands of cigarettes to their tar and nicotine content. CO TAR NICOTINE 15 1.2 16 15 1.2 16 17 1.0 16 6 0.8 9 1 0.1 1 8 0.8 8 10 0.8 10 17 1.0 16 15 1.2 15 11 0.7 9 18 1.4 18 16 1.0 15 10 0.8 9 7 0.5 5 18 1.1 16

Find the value of the linear correlation coefficient r. - 6 8 20 28 36 2 4 13 20 30

Construct a scatterplot and identify the mathematical model that best fits the data. Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use a calculator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several models and compare the values of R² -The table below shows the population of a city (in millions)in each year during the period 1990 - 1995. Using the number of years since 1990 as the independent variable, find the regression equation of the best model. 1990 1991 1992 1993 1994 1995 1.08 1.37 1.68 2.19 2.73 3.34

Is the data point, P -

Find the value of the linear correlation coefficient r. -Ten students in a graduate program were randomly selected. Their grade point averages (GPAs)when they entered the program were between 3.5 and 4.0. The following data were obtained regarding their GPAs on entering the program versus their current GPAs. Entering GPA Current GPA 3.5 3.6 3.8 3.7 3.6 3.9 3.6 3.6 3.5 3.9 3.9 3.8 4.0 3.7 3.9 3.9 3.5 3.8 3.7 4.0

In the context of regression, determine whether the following statement is true or false: If there is no correlation between x and y, the best predicted value of y for a given value of x is $\overline { \mathrm { y } }$

Construct a scatter diagram for the given data - 0.17 0.07 0.16 0.34 -0.16 0.3 0.51 0.08 0.5 0.85 0.35 0.46 0.01 0.53 0.22 -0.03

The equation of the regression line for the paired data below is $\hat { y } = 3 x$ . Find the unexplained variation. 2 4 5 6 7 11 13 20

Construct a scatterplot and identify the mathematical model that best fits the data. Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use a calculator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several models and compare the values of R² -A rock is dropped from a tall building and its distance (in feet)below the point of release is recorded as accurately as possible at various times after the moment of release. The results are shown in the table. Find the regression equation of the best model. (seconds after release) 1 2 3 4 5 6 (distance in feet) 16 63 146 255 403 572

The equation of the regression line for the paired data below is $\hat{\mathrm { y }} = 3 \mathrm { x }$ and the standard error of estimate is se $=$ 2.2361. Find the $90 \%$ prediction interval of $y$ for $x = 3$ . x 2 4 5 6 y 7 11 13 20

Find the value of the linear correlation coefficient r - 21.5 10.8 34.9 48.6 45.3 5 2 7 5 7

Describe the error in the stated conclusion. -Given: The linear correlation coefficient between scores on a math test and scores on a test of athletic ability is negative and close to zero. Conclusion: People who score high on the math test tend to score lower on the test of athletic ability.

The data set given below has $\hat { y } = 3 \mathrm { x } - 5$ for its regression equation. Find the standard error of estimate $s _ { \mathrm { e } }$ . x 3 2 5 8 y 4 1 10 19

Is the data point, P -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