Exam 16: Multiple Regression and Correlation

Nutritionist A nutritionist is analyzing the cost of an 8 oz.serving of pasta.The nutritionist anticipates that cost is related to: x₁ = Grams of protein/8 oz. x₂ = Grams of carbohydrates/8 oz. x₃ = Grams of fat/8 oz. Using MINITAB,the nutritionist obtained the following results: The regression equation is $Y = 1.39 + 0.0178 \times 1 - 0.0258 \times 2 - 0.00050 \times 3$ Predictor Coef Stdev t-ratio Constant 1.3928 0.1096 12.71 X1 0.017806 0.006600 2.70 X2 -0.025825 0.001613 -16.01 X3 -0.000501 0.002779 -0.18 $s=0.04805 \quad \mathrm{R}-\mathrm{sq}=97.5 \% \quad \mathrm{R}-\mathrm{sq}(\mathrm{adj})=96.6 \%$ Analysis of Variance SOURCE DF SS MS Regression 3 0.72562 0.24187 Error 8 0.01847 0.00231 Total 11 0.74409 -From these regression results,compute a 95% prediction interval for y when x₁ = 4,x₂ = 5,and x₃ = 3.

Some computer packages report an adjusted R² along with the unadjusted R².In general,why is the adjusted R² smaller,and under what conditions will it approach the size of the unadjusted R²?

A multiple regression model has three independent variables.The following values of y are given: 42 40 40 70 52 28 55 48 60 50 Compute the total sum of squares (SST). SST = ____________________

Consider the multiple regression equation, $\hat { Y }$ = 80 + 15x₁ - 5 x₂ + 100x₃.If x₁ = 10,x₂ = 4,x₃ = 12,what is the estimated value of y?

Grade A statistics teacher collected the following data to determine if the number of hours a student studied during the semester and the number of classes missed could be used to predict the final grade for the course.The following table shows the results of the model being applied to 8 students. Student Predicted Grade Actural Grade 1 84 92 2 93 95 3 77 81 4 81 78 5 81 75 6 89 88 7 88 85 85 85 -Calculate the coefficient of multiple determination.

In multiple regression and correlation analysis,what is the purpose of the regression component? Of the correlation component?

A multiple regression equation has been developed for y = daily attendance at a community swimming pool,x₁ = temperature (degrees Fahrenheit),and x₂ = weekend versus weekday,(x₂ = 1 for Saturday and Sunday,and 0 for other days of the week.)For the regression equation shown below,interpret each partial regression coefficient: $\hat{ Y }$ = 100 + 10x₁ + 175x₂.

Professor A statistics professor investigated some of the factors that affect an individual student's final grade in his course.He proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \epsilon$ where: y = final mark (out of 100) x₁ = number of lectures skipped x₂ = number of late assignments x₃ = mid-term test mark (out of 100) The professor recorded the data for 50 randomly selected students.The computer output is shown below. The regression equation is: $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predictor Coef StDev T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74$ $\mathrm { R } - \mathrm { Sq } = 30.0 \%$ Analysis of Variance Source of Variation df SS MS F Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 -Interpret the coefficients b₁ and b₃. b₁ = ____________________ Interpretation: _____________________________________________________ b₃ = ____________________ Interpretation: _____________________________________________________

Multicollinearity is a situation in which two or more of the independent variables are highly correlated with each other.

A multiple regression model has three independent variables.The following values of y and $\hat { Y }$ are given: y 40.60 39.40 38.80 70.60 49.50 25.50 52.30 46.40 56.60 47.40 47.07 44.50 41.09 50.16 54.52 36.25 53.67 53.51 40.20 46.14 Compute the multiple standard error of the estimate.

The purpose of the multiple correlation analysis is to measure the strength of the relationship between the dependent (y)and the set of independent (x)variables.

Multiple regression analysis examines the linear relationship between a dependent variable (y)and two or more independent variables (x₁,x₂,and so on).

Equation The regression equation, $\hat{ Y }$ = 4 + 1.5x₁ + 2.5x₂ has been fitted to 25 data points.The means of x₁ and x₂ are 30 and 46,respectively.The sum of the squared differences between observed and predicted values of y has been calculated as SSE = 175,and the sum of the squared differences between y values and mean of y is SST = 525. -What is the approximate 95% prediction interval for an individual y whenever x₁ = 20 and x₂ = 25?

Professor A statistics professor investigated some of the factors that affect an individual student's final grade in his course.He proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \epsilon$ where: y = final mark (out of 100) x₁ = number of lectures skipped x₂ = number of late assignments x₃ = mid-term test mark (out of 100) The professor recorded the data for 50 randomly selected students.The computer output is shown below. The regression equation is: $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predictor Coef StDev T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74$ $\mathrm { R } - \mathrm { Sq } = 30.0 \%$ Analysis of Variance Source of Variation df SS MS F Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 -Do these data provide enough evidence at the 1% significance level to conclude that the final mark and the mid-term mark are positively linearly related? Test statistic = ____________________ Critical Value = ____________________ Conclusion: ____________________

Marketing Analyst A marketing analyst is interested in predicting prospective buyer's knowledge about compact disc players.A random sample of 36 buyers was taken,a questionnaire about compact disc players completed,and information about education,income and age was obtained.In estimating the equation,the variables were: y = knowledge about compact disc players x₁ = education (years) x₂ = age x₃ = income (thousands of dollars) The resulting output using MINITAB was: The regression equation is Y=50.2+4.36\times1-0.632\times2-0.068\times3 Predictor Coef Stdev t-ratio Constant 50.168 4.977 10.08 X1 4.3609 0.4064 10.73 X2 -0.63169 0.08172 -7.73 X3 -0.0682 0.1176 -0.58 s=4.615 R-sq =85.0\% R-sq(adj) =83.6\% -Identify b₀,b₁,and b₃. b₀ = ____________________ b₁ = ____________________ b₃ = ____________________

What does the numerical value of the multiple standard error of estimate (s_e)reflect?

Consider the multiple regression equation, $\hat{ Y }$ = 80 + 15x₁ - 5x₂ + 100x₃.If x₃ were to increase by 5,what change would be necessary in x₂ in order for the estimated value of y to remain unchanged? x₂ would ____________________ by ____________________.

With four or more variables,the regression equation becomes a mathematical entity called a ____________________.

Motor Vehicle In order to predict motor vehicle purchases for the U.S.,the coefficients of a multiple regression equation were estimated using 25 years of data.The variables were: y = motor vehicle purchases (billions of dollars) x₁ = disposable personal income (billions of dollars) x₂ = U.S.population (millions of persons) x₃ = automobile installment credit (billions of dollars) Part of the results using MINITAB was: The regression equation is $Y = - 61.3 - 0.0022 \times 1 + 0.368 \times 2 + 0.725 \times 3$ Predictor Coef Stdev t-ratio Constant -61.28 36.96 -1.66 X1 -0.00221 0.01504 -0.15 X2 0.3679 0.2001 1.84 X3 0.7254 0.2115 3.43 Analysis of Variance SOURCE DF SS Regression 3 32624 Error 21 591 Total 24 33215 Use the values in the analysis of variance table to find MSR and MSE. MSR = ____________________ MSE = ____________________

Nutritionist A nutritionist is analyzing the cost of an 8 oz.serving of pasta.The nutritionist anticipates that cost is related to: x₁ = Grams of protein/8 oz. x₂ = Grams of carbohydrates/8 oz. x₃ = Grams of fat/8 oz. Using MINITAB,the nutritionist obtained the following results: The regression equation is $Y = 1.39 + 0.0178 \times 1 - 0.0258 \times 2 - 0.00050 \times 3$ Predictor Coef Stdev t-ratio Constant 1.3928 0.1096 12.71 X1 0.017806 0.006600 2.70 X2 -0.025825 0.001613 -16.01 X3 -0.000501 0.002779 -0.18 $s=0.04805 \quad \mathrm{R}-\mathrm{sq}=97.5 \% \quad \mathrm{R}-\mathrm{sq}(\mathrm{adj})=96.6 \%$ Analysis of Variance SOURCE DF SS MS Regression 3 0.72562 0.24187 Error 8 0.01847 0.00231 Total 11 0.74409 -From these regression results,compute a 95% confidence interval for y when x₁ = 4,x₂ = 5,and x₃ = 3.