Exam 10: Correlation and Regression

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 value(s) of $r$ given that $n = 10$ and $\alpha = 0.05$ .

Use computer software to find the multiple regression equation. Can the equation be used for prediction? A wildlife analyst gathered the data in the table to develop an equation to predict The weights of bears. He used WEIGHT as the dependent variable and CHEST, LENGTH, And SEX as the independent variables. For SEX, he used male=1 and female=2. WEIGHT CHEST LENGTH SEX 344 45.0 67.5 1 416 54.0 72.0 1 220 41.0 70.0 2 360 49.0 68.5 1 332 44.0 73.0 1 140 32.0 63.0 2 436 48.0 72.0 1 132 33.0 61.0 2 356 48.0 64.0 2 150 35.0 59.0 1 202 40.0 63.0 2 365 50.0 70.5 1

The table below lists weights (carats) and prices (dollars) for randomly selected diamonds. Find the regression equation, letting the weight be the predictor variable. Find the best predicted price for a diamond with a weight of $1.50$ carats. What is wrong with predicting the price of a $1.50$ -carat diamond? Weight 0.3 0.4 0.5 0.5 1.0 0.7 Price 510 1151 1343 1410 5669 2277

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: $\hat { y } = 31.6 + 10.9 x$ , where $x$ represents the number of years of study and $y$ represents the grade on the test. Identify the predictor and response variables.

Determine which plot shows the strongest linear correlation.

Use computer software to find the best multiple regression equation to explain the variation in the dependent variable, $Y$ , in terms of the independent variables, $X _ { 1 } , X _ { 2 } , X _ { 3 }$ . Y 98.6 87.4 108.5 101.2 97.6 110.1 102.4 96.7 110.4 CORRELATION COEFFICIENTS 100.9 98.2 104.3 102.3 99.8 107.2 Y/ =0.850 101.5 100.5 105.8 YI =0.742 101.6 103.2 107.8 101.6 107.8 103.4 99.8 96.6 102.7 COEFFICIENT OF DETERMINATION 100.3 88.9 104.1 Y/ =0.723 97.6 75.1 99.2 Y/ =0.550 97.2 76.9 99.7 Yl ,=0.867 97.3 84.6 102.0 96.0 90.6 94.3 99.2 103.1 97.7 100.3 105.1 101.1 100.3 96.4 102.3 104.1 104.4 104.4 105.3 110.7 108.5 107.6 127.1 111.3

Find the indicated multiple regression equation. Below are performance and attitude ratings of employees. Performance 59 63 65 69 58 77 76 69 70 64 Attitude 72 67 78 82 75 87 92 83 87 78 Managers also rate the same employees according to adaptability, and below are the results that correspond to those given above. start text Adaptability colon end text 50 52 54 60 46 67 66 59 62 55 Find the multiple regression equation that expresses performance in terms of attitude and adaptability.

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 ( ) 43 53 59 42 47 53 55 63 67 75 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 dexterity scores.

Use computer software to find the best multiple regression equation to explain the variation in the dependent variable, $Y$ , in terms of the independent variables, $X _ { 1 } , X _ { 2 } , X _ { 3 }$ . Y 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 CORRELATION COEFFICIENT Y/=0.886 Y/=0.965 COEFFICIENTS OF DETERMINATION Y/=0.932 Y/,=0.943

Applicants for a particular job, which involves extensive travel in Spanish speaking countries must take a proficiency test in Spanish. The sample data below were obtained in study of the relationship between the numbers of years applicants have studied Spanish ( $x )$ and their score on the test $( y )$ . x 3 4 4 2 5 3 4 5 3 2 y 57 78 72 58 89 63 73 84 75 48 The regression equation $\hat { y } = 31.55 + 10.90 x$ was obtained. Construct a residual plot for the data.

Use computer software to obtain the multiple regression equation and identify $R ^ { 2 }$ , adjusted $R ^ { 2 }$ , and the $P$ -value. An anti-smoking group used data in the table to relate the carbon monoxide (CO) of various brands of cigarettes to their tar and nicotine (NIC) content. CO TAR NIC 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

Use computer software to find the multiple regression equation. Can the equation be used for prediction? An anti-smoking group used data in the table to relate the carbon monoxide( CO) Of various brands of cigarettes to their tar and nicotine (NIC)content. CO TAR NIC 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

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

When testing to determine if correlation is significant, we use the hypotheses $H _ { 0 } : \rho = 0$ , $H _ { 1 } : \rho \neq 0$ . What does the symbol $\rho$ represent? Explain the meaning of the null and alternative hypotheses.

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. Two different tests are designed to measure employee Productivity and dexterity. Several employees are randomly selected and tested with these Results. Productivity (x) 23 25 28 21 21 25 26 30 34 36 Dexterity (y) 49 53 59 42 47 53 55 63 67 75

Find the value of the linear correlation coefficient r. The paired data below consist of the costs of advertising (in thousands of dollars)and the number of products sold (in thousands). Cost 9 2 3 4 2 5 9 10 Number 85 52 55 68 67 86 83 73

Sketch an example of a residual plot that suggests that a regression equation is not a good model. Be sure to include at least 10 points in your example.

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 = 0.543, n = 25

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 $0.05$ . $r = - 0.816 , n = 5$

Find the explained variation for the paired data. The equation of the regression line for the paired data below is $\hat { y } = 6.18286 + 4.33937 x$ . 9 7 2 3 4 22 17 43 35 16 21 23 102 81