Exam 10: Correlation and Regression

Solve the problem. -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 $E = \left( t _ { \alpha / 2 } \right) \operatorname { se } \sqrt { \frac { 1 } { n } + x ^ { 2 } / \left[ \sum x ^ { 2 } - \left( \sum x \right) ^ { 2 } / n \right] }$ 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 way from the sample data. Use the data below to obtain a $95 \%$ confidence interval estimate of $\beta _ { 0 }$ . x (hours studied) 2.5 4.5 5.1 7.9 11.6 y (score on test) 66 70 60 83 93

Construct the indicated prediction interval for an individual y. -The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\hat { \mathrm { y } } = 44.845 + 3.524 \mathrm { x }$ and the standard error of estimate is se $= 5.40$ . Find the $99 \%$ prediction interval for the test score of a person who spent 7 hours preparing for the test. x Hours of preparation 5 2 9 6 10 Test score 64 48 72 73 80

Use the computer display to answer the question. -A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear Regression line and the computer output is shown below. Along with the paired sample data, the program was Also given an x value of 2 (years of study) to be used for predicting test score. The regression equation is Score $= 31.55 + 10.90$ Years. $\mathrm { S } = 5.651 \quad \mathrm { R } - \mathrm { Sq } = 83.0 \% \quad \mathrm { R } - \mathrm { Sq } ($ Adj $) = 82.7 \%$ Predicted values Fit StDev Fit 95.0\% CI 95.0\% PI 53.35 3.168 (42.72,63.98) (31.61,75.09) For a person who studies for 2 years, obtain the 95% prediction interval and write a statement interpreting the Interval.

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 $\alpha$ -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 $x = 5$ ?

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 $\mathrm { R } ^ { 2 }$ - x 1 2 3 4 5 6 y 9 13 25 27 31 46

The following table gives the US domestic oil production rates (excluding Alaska) from 1987 to 2002. A regression equation was fit to the data and the residual plot is shown below. Does the residual plot suggest that the regression equation is a bad model? Why or why not?

Find the indicated multiple regression equation. -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 following 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 C in terms of Test A and Test B.

Nine adults were selected at random from among those working full time in the town of Workington. Each person was asked the number of years of college education they had completed and was also asked to rate their job satisfaction on a scale of 1 to 10. The pairs of data values area plotted in the scatterplot below. The four points in the lower left corner correspond to employees from company A and the five points in the upper right corner correspond to employees from company B. a. Using the pairs of values for all 9 points, find the equation of the regression line. b. Using only the pairs of values for the four points in the lower left corner, find the equation of the regression line. c. Using only the pairs of values for the five points in the upper right corner, find the equation of the regression line. d. Compare the results from parts a, b, and c.

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

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

Construct the indicated prediction interval for an individual y. -The paired data below consists of heights and weights of 6 randomly selected adults. The equation of the regression line is $\hat { y } = - 181.342 + 144.46 x$ and the standard error of estimate is $\mathrm { s } _ { \mathrm { e } } = 5.0015$ . Find the $95 \%$ prediction interval for the weight of a person whose height is $1.75 \mathrm {~m}$ . x Height (meters) 1.61 1.72 1.78 1.80 1.67 1.88 y Weight () 54 62 70 84 61 92

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 $\alpha$ - $\mathrm { n } = 45 , \alpha = 0.01$

Use computer software to obtain the multiple regression equation and identify R $\mathrm { R } ^ { 2 }$ , adjusted R $\mathrm { R } ^ { 2 }$ , 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 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

Find the indicated multiple regression equation. -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 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, $\mathrm { X } _ { 1 } , \mathrm { X } _ { 2 } , \mathrm { X } _ { 3 }$ -

Construct the indicated prediction interval for an individual y. -The regression equation for the given paired data is $\hat { y } = 73.012 + 0.8859 x$ and the standard error of estimate is $\mathrm { s_e } = 2.78807 .$ Find the $95 \%$ prediction interval of $\mathrm { y }$ for $\mathrm { x } = 37$ . 25 26 36 36 40 48 95 95 102 109 110 114

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. - 24 26 28 30 32 15 13 20 16 24

Is the data point, P, an outlier, an influential point, both, or neither? -

The following residual plot is obtained after a regression equation is determined for a set of data. Does the residual plot suggest that the regression equation is a bad model? Why or why not?

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