Exam 10: Correlation and Regression

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. Is there sufficient evidence to suggest that there is a linear correlation between weights and prices? Construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r using $\alpha = 0.05$ Weight 0.3 0.4 0.5 0.5 1.0 0.7 Price 510 1151 1343 1410 5669 2277

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

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, $\mathrm { X } _ { 1 } , \mathrm { X } _ { 2 } , \mathrm { X } _ { 3 }$ - 344 45.0 67.5 1 CORRELATION COEFFICIENTS 416 54.0 72.0 1 /=0.951 220 41.0 70.0 2 /=0.789 360 49.0 68.5 1 /=-0.616 332 44.0 73.0 1 140 32.0 63.0 2 436 48.0 72.0 1 132 33.0 61.0 2 =0.905 356 48.0 64.0 2 ,=0.919 150 35.0 59.0 1 ,,=0.927 202 40.0 63.0 2 365 50.0 70.5 1

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 $\text { Score }=31.55+10.90 \text { Years. }$ Predictor Coef StDev Constant 31.55 6.360 4.96 0.000 Years 10.90 1.744 6.25 0.000 $\mathrm{S}=5.651 \quad \mathrm{R}-\mathrm{Sq}=83.0 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{Adj})=82.7 \%$ $\text { Predicted values }$ Fit StDev Fit 95.0\% CI 95.0\% PI 53.35 3.168 (42.72,63.98) (31.61,75.09) What percentage of the total variation in test scores can be explained by the linear relationship between years of Study and test scores?

The results for several randomly selected students for test 1 and test 2 grades are given below. Test 1 59 63 65 69 58 77 76 69 70 64 Test 2 72 67 78 82 75 87 92 83 87 78 Is there sufficient evidence to suggest that there is a linear correlation between test 1 and test 2 grades? Construct a scatterplot, and find the value of the linear correlation coefficient r. Also, find the P-value or the critical value(s) of r using $\alpha = 0.05$

Find the unexplained variation for the paired data. The equation of the regression line for the paired data below is $y = 3 x$ 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 $\mathrm { R } ^ { 2 }$ - 1 2 3 4 5 7 17 20 25 28

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, $\mathrm { X } _ { 1 } , \mathrm { X } _ { 2 } , \mathrm { X } _ { 3 }$ -

The table lists the value y (in dollars) of $100 deposited in a certificate of deposit (CD) at a bank after y years. Year 1 2 3 4 5 20 Value 103 106.09 109.27 112.55 115.93 180.61 Construct a scatterplot and identify the mathematical model that best fits the given 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. Include the type of model and the equation for the model you find.

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. 6 8 20 28 36 2 4 13 20 30

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 $\alpha$ -Nine pairs of data yield $\mathrm { r } = 0.867$ and the regression equation $\hat { \mathrm { y } } = 19.4 + 0.93 \mathrm { x } .$ Also, $\overline { \mathrm { y } } = 64.7$ . What is the best predicted value of $y$ for $x = 44$ ?

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?

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 = 15 and $\alpha = 0.05$

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 $\alpha$ -Given: There is no significant linear correlation between scores on a math test and scores on a verbal test. Conclusion: There is no relationship between scores on the math test and scores on the verbal test.

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

Describe what scatterplots are and discuss the importance of creating scatterplots.

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 $\alpha$ -Eight pairs of data yield $\mathrm { r } = 0.742$ and the regression equation $\hat { y } = 55.8 + 2.79 x .$ Also, $\bar { y } = 71.125$ . What is the best predicted value of $y$ for $x = 8$ ?

Give an example of a pair of variables which you would expect to have a negative linear correlation coefficient and explain why.