Exam 10: Correlation and Regression

Find the best predicted value of y corresponding to the given value of x. -Four pairs of data yield $\mathrm { r } = 0.942$ and the regression equation $\mathrm { y } = 3 \mathrm { x }$ . Also, $\overline { \mathrm { y } } = 12.75$ . What is the best predicted value of $y$ for $x = 2.8$ ?

Determine which plot shows the strongest linear correlation

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

Provide an appropriate response. -Define the term independent, or predictor, variable and the term dependent, or response, variable. Give examples for each.

Find the best predicted value of y corresponding to the given value of x. -The regression equation relating attitude rating $( \mathrm { x } )$ and job performance rating $( \mathrm { y } )$ for the employees of a company is $\hat{y} = 11.7 + 1.02 x$ . Ten pairs of data were used to obtain the equation. The same data yield $r = 0.863$ and $\bar { y } = 80.1$ . What is the best predicted job performance rating for a person whose attitude rating is 73 ?

The paired data below consists of heights and weights of 6 randomly selected adults. The equation of the regression line is $\hat { y } = - 181.342 + 144.46 x$ . Find the total variation. x Height (meters) 1.61 1.72 1.78 1.80 1.67 1.88 Weight () 54 62 70 84 61 92

The paired data below consists of heights and weights of 6 randomly selected adults. The equation of the regression line is $\hat{\mathrm { y }} = - 181.342 + 144.46 \mathrm { x }$ and the standard error of estimate is $\mathrm { se } = 5.0015$ . 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 Weight () 54 62 70 84 61 92

Provide an appropriate response. -Create a scatterplot that shows a perfect positive correlation between x and y. How would the scatterplot change if the correlation showed a)a strong positive correlation, b)a positive correlation, and c)no correlation?

Use computer software to obtain the regression and identify R², adjusted R², and the P-value -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

A regression equation is obtained for a collection of paired data. It is found that the total variation is 114, the explained variation is 91.7, and the unexplained variation is 22.3. Find the coefficient of determination.

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 6 9 13 25 27 31 46

Provide an appropriate response. -Describe what correlation is, and explain the purpose of correlation.

Find the value of the linear correlation coefficient r - 62 53 64 52 52 54 58 y 158 176 151 164 164 174 162

Is the data point, P -

Use computer software to obtain the regression and identify R², adjusted R², and the P-value -FPEA, the Farm Production Enhancement Agency, regressed corn output against acreage, rainfall, and a trend line. The trend line is proxy for technological advancement in farming from improved pest control, fertilization, land management, and farming implements. CORNPROD ACRES RAINFALL TREND 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

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. Predictor Coef StDev T P Constant 31.55 6.360 4.96 0.000 Years 10.90 1.744 6.25 0.000 =5.651 -=83.0\% -()=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) What percentage of the total variation in test scores is unexplained by the linear relationship between years of study and test scores?

Find the value of the linear correlation coefficient r. - 3 5 7 15 16 y 8 11 7 14 20

The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\hat{\mathrm { y }} = 44.8447 + 3.52427 \mathrm { x }$ . Find the standard error of estimate. x Hours of preparation 5 2 9 6 10 y Test score 64 48 72 73 80

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 1 1.2 2.7 4.4 6.6 9.5 1.6 4.7 8.9 9.5 12.0

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. Predictor Coef StDev T P Constant 31.55 6.360 4.96 0.000 Years 10.90 1.744 6.25 0.000 $\mathrm{S}=5.651 \quad \mathrm{R}-\mathrm{Sq}=83.0 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{Adj})=82.7 \%$ 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 in