Exam 10: Correlation and Regression

The test scores of 6 randomly picked students and the numbers of hours they prepared are as follows: Hours 5 10 4 6 10 9 Score 64 86 69 86 59 87 The equation of the regression line is $\hat { y } = 1.06604 \mathrm { x } + 67.3491$ . Find the coefficient of determination.

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

The equation of the regression line for the paired data below is $\hat { y } = 6.18286 + 4.33937 x$ . Find the unexplained variation. 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 -A visitor to Yellowstone National Park sat down one day and observed Old Faithful, which faithfully spurts throughout the day, day in and day out. He surmised that the height of a given spurt was caused by the pressure build-up during the interval between spurts and by the momentum build-up during the duration of the spurt. He wrote down the data to test his hypothesis, but he didn't know what to do with his data. Can you help him out with his theory? HEIGHT INTERVAL DURATIO 150 86 240 154 86 237 140 62 122 140 104 267 160 62 113 140 95 258 150 79 232 150 62 105 160 94 276 155 79 248 125 86 243 136 85 241 140 86 214 155 58 114 130 89 272 125 79 227 125 83 237 139 82 238 125 84 203 140 82 270 140 82 270 140 78 218 135 87 270 140 70 241 100 56 102 105 81 271

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

A regression equation is obtained for a collection of paired data. It is found that the total variation is 25.753, the explained variation is 18.658, and the unexplained variation is 7.095. Find the coefficient of determination.

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 th given above. Adaptability: 50 52 54 60 46 67 66 59 62 55 Find the multiple regression equation that expresses performance in terms of attitude and adaptability.

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 - 1 2 3 4 5 7 17 20 25 28

Construct a scatter diagram for the given data - -4 3 7 5 11 10 6 -1 -1 2 7 7 9 9 6 7 2 3

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

A fitness rating was obtained for 9 randomly selected adult women. Each person was also asked her age, weight, and the number of hours she spent exercising each week. The results are shown below. Age 39 27 41 48 56 59 22 64 35 Weight 140 129 137 125 162 152 118 142 126 Hours of exercise per week 2 6 4 9 0 3 11 3 4 Fitness rating 72 88 63 84 47 52 90 31 64 Identify the multiple regression equation that expresses fitness in terms of age, weight, and hours of exercise per week.

In the context of regression, determine whether the following statement is true or false: If there is a very strong correlation between x and y, the amount of unexplained variation should be relatively large.

A confidence interval for the $y$ -intercept $\beta _ { 0 }$ for a regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ can be found by evaluating the limits in the interval below: $\mathrm { b } _ { 0 } - \mathrm { E } < \beta _ { 0 } < \mathrm { b } _ { 0 } + \mathrm { E } ,$ where $\mathrm { E } = \left( \mathrm { t } _ { \alpha / 2 } \right) \mathrm { se } \sqrt { \frac { 1 } { \mathrm { n } } + \overline { \mathrm { x } } 2 / \left[ \sum \mathrm { x } ^ { 2 } - \left( \sum \mathrm { x } \right) ^ { 2 } / \mathrm { n } \right] }$ . The critical value $t _ { \alpha / 2 }$ is found from the $t$ -table using $n - 2$ degrees of freedom and $b _ { 0 }$ is calculated in the usual v from the sample data. Use the data below to obtain a $95 \%$ confidence interval estimate of $\beta _ { 0 }$ . (hours studied) 2.5 4.5 5.1 7.9 11.6 (score on test) 66 70 60 83 93

Find the best predicted value of y corresponding to the given value of x. -The regression equation relating dexterity scores $( \mathrm { x } )$ and productivity scores ( $\mathrm { y } )$ for the employees of a company $\hat { y } = 5.50 + 1.91 \mathrm { x }$ . Ten pairs of data were used to obtain the equation. The same data yield $\mathrm { r } = 0.986$ and $\bar { y } = 56.3$ . What is the best predicted productivity score for a person whose dexterity score is 33 ?

Find the value of the linear correlation coefficient r. - 1.2 1.4 1.6 1.8 2.0 54 53 55 54 56

Below are the results of two separate tests designed to measure a student's ability to solve problems. Test A 48 52 58 44 43 43 40 51 59 Test B 73 67 73 59 58 56 58 64 74 In addition to these results, a third test was designed to measure the same problem-solving ability, and the follo' results correspond to the same students. Test C: 48 41 59 45 42 44 40 58 60 Find the multiple regression equation that expresses results from Test $\mathrm { C }$ in terms of Test $\mathrm { A }$ and Test $\mathrm { B }$ .

Find the best predicted value of y corresponding to the given value of x. -Ten pairs of data yield $\mathrm { r } = 0.003$ and the regression equation $\hat { \mathrm { y } } = 2 + 3 \mathrm { x }$ . Also, $\overline { \mathrm { y } } = 5.0$ . What is the best predicted value of $y$ for $x = 2$ ?

Use computer software to find the best regression equation to explain the variation in the dependent variable, Y, in terms of the independent variables, X1, X2, X3 - Y 456 9896 29.1 1 421 9680 42.3 2 653 10449 29.8 3 573 10811 26.0 4 546 10014 34.3 5 499 10293 22.7 6 504 9413 24.2 7 611 9860 31.6 8 646 9782 25.6 9 789 12139 37.9 10 773 12166 33.9 11 753 9976 37.4 12 852 10645 27.0 13 755 9738 31.5 14 815 9933 39.9 15 902 10132 25.3 16 986 11145 30.4 17 909 9775 32.7 18 945 9549 35.0 19 866 10077 33.8 20 1178 11550 29.4 21 1230 10600 37.1 22 1207 11280 42.9 23 968 12100 32.2 24 1118 12420 30.5 25 CORRELATION COEFFICIENTS /=.509 /=.280 /=.930 COEFFICIENTS OF DETERMINATION /=.259 /=.079 /=.864 /,=.880 /,,=.884

Provide an appropriate response. -Suppose there is significant correlation between two variables. Describe two cases under which it might be inappropriate to use the linear regression equation for prediction. Give examples to support these cases.

Find the best predicted value of y corresponding to the given value of x. -Six pairs of data yield $\mathrm { r } = 0.444$ and the regression equation $\hat { \mathrm { y } } = 5 \mathrm { x } + 2$ . Also, $\overline { \mathrm { y } } = 18.3$ . What is the best predicted value of $y$ for $x = 5$ ?