Exam 10: Correlation and Regression

Construct the indicated prediction interval for an individual y. -The equation of the regression line for the paired data below is $\hat { y } = 6.1829 + 4.3394 x$ and the standard error of estimate is $\mathrm { se } ^ { = } = 1.6419$ . Find the $99 \%$ prediction interval of $\mathrm { y }$ for $\mathrm { x } = 6$ . 9 7 2 3 4 22 17 43 35 16 21 23 102 81

The following table gives the US domestic oil production rates (excluding Alaska) over the past few years. 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?

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. $r = 0.767 , n = 25$

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. -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

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. - 1.2 1.4 1.6 1.8 2.0 54 53 55 54 56

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 $\alpha$ -Four pairs of data yield $\mathrm { r } = 0.942$ and the regression equation $\hat { \mathrm { y } } = 3 \mathrm { x }$ . Also, $\overline { \mathrm { y } } = 12.75$ . What is the best predicted value of $y$ for $x = 4.6$ ?

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 $\alpha$ -Based on the data from six students, the regression equation relating number of hours of preparation (x) and test score $( \hat { y } )$ is $\mathrm { y } = 67.3 + 1.07 \mathrm { x }$ . The same data yield $\mathrm { r } = 0.224$ and $\overline { \mathrm { y } } = 75.2$ . What is the best predicted test score for a student who spent 2 hours preparing for the test?

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. 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.

Determine which scatterplot shows the strongest linear correlation. -Which shows the strongest linear correlation?

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 $\mathrm { R } ^ { 2 }$ -The sum of squares of residuals $\sum ( y - \hat { y } ) ^ { 2 }$ can be used to assess the quality of a regression model. A residual is the difference between an observed y value and the value of y predicted from the model, $\hat { y }$ . The better the model, the smaller the sum of squares of residuals. For the data below find the sum of squares of residuals which results from fitting a linear model, and the sum of squares of residuals which results from fitting a logarithmic model. Which model fits better? How can you tell? x 1 2 3 4 5 y 7 17 20 25 28

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

Find the explained variation for the paired data. -The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\hat { y } = 44.8447 + 3.52427 x$ . Find the explained variation. x Hours of preparation 5 2 9 6 10 y Test of 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 $\mathrm { S } = 5.651 \quad \mathrm { R } - \mathrm { Sq } = 83.0 \% \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { Adj } ) = 82.7 \%$ Predicted values Fit StDev Fit 95.0\% CI 95.0\% PI 53.35 3.168 (42.72,63.98) (31.61,75.09) Use the information in the display to find the value of the linear correlation coefficient r. Determine whether There is significant linear correlation between years of study and test scores. Use a significance level of 0.05. There are 10 pairs of data.

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

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

Determine which plot shows the strongest linear correlation.

Is the data point, P, an outlier, an influential point, both, or neither? -The regression equation for a set of paired data is $\hat { y } = 67.1 + - 0.9 x$ . The values of $x$ run from 100 to 400 . A new data point, $\mathrm { P } ( 399 , - 292 )$ , is added to the set.

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

Provide an appropriate response. For the data below, determine the logarithmic equation, $y = a + b \ln x$ that best fits the data. Hint: Begin by replacing each x-value with ln x then use the usual methods to find the equation of The least squares regression line. x 1.2 2.7 4.4 6.6 9.5 y 1.6 4.7 8.9 9.5 12.0

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. - x 1 3 5 7 9 y 143 116 100 98 90