Bus Stat Ch. 14 (Part 2)
Given below are four observations collected in a regression study on two variables x (independent variable) and y (dependent variable). x y 2 4 6 7 9 8 9 9 a. Develop the least squares estimated regression equation. b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. c. Perform an F test to determine whether or not the model is significant. Let α = 0.05. d. Compute the coefficient of determination.
ANS: Regression Statistics Multiple R 0.977 R Square 0.955 Adjusted R Square 0.932 Standard Error 0.564 Observations 4 ANOVA df SS MS F Significance F Regression 1 13.364 13.364 42.000 0.023 Residual 2 0.636 0.318 Total 3 14 Coefficients Standard Error t Stat P-value Intercept 2.864 0.698 4.104 0.055 X 0.636 0.098 6.481 0.023 a. = 2.864 + 0.636x b. p-value < .05; reject Ho c. p-value < .05; reject Ho d. 0.955
Part of an Excel output relating X (independent variable) and Y (dependent variable) is shown below. Fill in all the blanks marked with "?". Summary Output Regression Statistics Multiple R 0.1347 R Square ? Adjusted R Square ? Standard Error 3.3838 Observations ? ANOVA df SS MS F Significance F Regression ? 2.7500 ? ? 0.632 Residual ? ? 11.45 Total 14 ? Coefficients Standard Error t Stat P-value Intercept 8.6 2.2197 ? 0.0019 x 0.25 0.5101 ? 0.632
ANS: Summary Output Regression Statistics Multiple R 0.1347 R Square 0.0181 Adjusted R Square -0.0574 Standard Error 3.384 Observations 15 ANOVA df SS MS F Significance F Regression 1 2.750 2.75 0.2402 0.6322 Residual 13 148.850 11.45 Total 14 151.600 Coefficients Standard Error t Stat p-value Intercept 8.6 2.2197 3.8744 0.0019 x 0.25 0.5101 0.4901 0.6322
Part of an Excel output relating X (independent variable) and Y (dependent variable) is shown below. Fill in all the blanks marked with "?". Summary Output Regression Statistics Multiple R ? R Square 0.5149 Adjusted R Square ? Standard Error 7.3413 Observations 11 ANOVA df SS MS F Significance F Regression ? ? ? ? 0.0129 Residual ? ? ? Total ? 1000.0000 Coefficients Standard Error t Stat P-value Intercept ? 29.4818 3.7946 0.0043 x ? 0.7000 -3.0911 0.0129
ANS: Summary Output Regression Statistics Multiple R 0.7176 R Square 0.5149 Adjusted R Square 0.4611 Standard Error 7.3413 Observations 11 ANOVA df SS MS F Significance F Regression 1 514.9455 514.9455 9.5546 0.0129 Residual 9 485.0545 53.8949 Total 10 1000.0000 Coefficients Standard Error t Stat P-value Intercept 111.8727 29.4818 3.7946 0.0043 x -2.1636 0.7000 -3.0911 0.0129
Shown below is a portion of a computer output for a regression analysis relating Y (dependent variable) and X (independent variable). ANOVA df SS Regression 1 115.064 Residual 13 82.936 Total Coefficients Standard Error Intercept 15.532 1.457 x -1.106 0.261 a. Perform a t test using the p-value approach and determine whether or not Y and X are related. Let α = 0.05. b. Using the p-value approach, perform an F test and determine whether or not X and Y are related. c. Compute the coefficient of determination and fully interpret its meaning. Be very specific.
ANS: a and b Summary Output Regression Statistics Multiple R 0.7623 R Square 0.5811 Adjusted R Square 0.5489 Standard Error 2.5258 Observations 15 ANOVA df SS MS F Significance F Regression 1 115.064 115.064 18.036 0.001 Residual 13 82.936 6.380 Total 14 198 Coefficients Standard Error t Stat P-value Intercept 15.532 1.457 10.662 0.000 x -1.106 0.261 -4.247 0.001 c. 58.11% of the variability in Y is explained by the variability in X.
Shown below is a portion of a computer output for a regression analysis relating Y (demand) and X (unit price). ANOVA df SS Regression 1 5048.818 Residual 46 3132.661 Total 47 8181.479 Coefficients Standard Error Intercept 80.390 3.102 X -2.137 0.248 a. Perform a t test and determine whether or not demand and unit price are related. Let α = 0.05. b. Perform an F test and determine whether or not demand and unit price are related. Let α = 0.05. c. Compute the coefficient of determination and fully interpret its meaning. Be very specific. d. Compute the coefficient of correlation and explain the relationship between demand and unit price.
ANS: a and b Summary Output Regression Statistics Multiple R 0.786 R Square 0.617 Adjusted R Square 0.609 Standard Error 8.252 Observations 48 ANOVA df SS MS F Significance F Regression 1 5048.818 5048.818 74.137 0.000 Residual 46 3132.661 68.101 Total 47 8181.479 Coefficients Standard Error t Stat P-value Intercept 80.390 3.102 25.916 0.000 X -2.137 0.248 -8.610 0.000 c. R2 = 0.617; 61.7% of the variability in demand is explained by the variability in price. d. R = -0.786; since the slope is negative, the coefficient of correlation is also negative, indicating that as unit price increases demand decreases.
Shown below is a portion of a computer output for a regression analysis relating supply (Y in thousands of units) and unit price (X in thousands of dollars). ANOVA df SS Regression 1 354.689 Residual 39 7035.262 Coefficients Standard Error Intercept 54.076 2.358 X 0.029 0.021 a. What has been the sample size for this problem? b. Perform a t test and determine whether or not supply and unit price are related. Let α = 0.05. c. Perform and F test and determine whether or not supply and unit price are related. Let α = 0.05. d. Compute the coefficient of determination and fully interpret its meaning. Be very specific. e. Compute the coefficient of correlation and explain the relationship between supply and unit price. f. Predict the supply (in units) when the unit price is $50,000.
ANS: a through c Regression Statistics Multiple R 0.219 R Square 0.048 Adjusted R Square 0.024 Standard Error 13.431 Observations 41 ANOVA df SS MS F Significance F Regression 1 354.689 354.689 1.966 0.169 Residual 39 7035.262 180.391 Total 40 7389.951 Coefficients Standard Error t Stat P-value Intercept 54.076 2.358 22.938 0.000 X 0.029 0.021 1.402 0.169 d. R2 = 0.048; 4.8% of the variability in supply is explained by the variability in price. e. R = 0.219; since the slope is positive, as unit price increases so does supply. f. supply = 54.076 + .029(50) = 55.526 (55,526 units)
Shown below is a portion of a computer output for regression analysis relating Y (dependent variable) and X (independent variable). ANOVA df SS Regression 1 24.011 Residual 8 67.989 Coefficients Standard Error Intercept 11.065 2.043 x -0.511 0.304 a. What has been the sample size for the above? b. Perform a t test and determine whether or not X and Y are related. Let α = 0.05. c. Perform an F test and determine whether or not X and Y are related. Let α = 0.05. d. Compute the coefficient of determination. e. Interpret the meaning of the value of the coefficient of determination that you found in d. Be very specific.
ANS: a through d Summary Output Regression Statistics Multiple R 0.511 R Square 0.261 Adjusted R Square 0.169 Standard Error 2.915 Observations 10 ANOVA df SS MS F Significance F Regression 1 24.011 24.011 2.825 0.131 Residual 8 67.989 8.499 Total 9 92 Coefficients Standard Error t Stat P-value Intercept 11.065 2.043 5.415 0.001 x -0.511 0.304 -1.681 0.131 e. 26.1% of the variability in Y is explained by the variability in X.
Shown below is a portion of an Excel output for regression analysis relating Y (dependent variable) and X (independent variable). ANOVA df SS Regression 1 110 Residual 8 74 Total 9 184 Coefficients Standard Error Intercept 39.222 5.943 x -0.5556 0.1611 a. What has been the sample size for the above? b. Perform a t test and determine whether or not X and Y are related. Let α = 0.05. c. Perform an F test and determine whether or not X and Y are related. Let α = 0.05. d. Compute the coefficient of determination. e. Interpret the meaning of the value of the coefficient of determination that you found in d. Be very specific.
ANS: a through d Summary Output Regression Statistics Multiple R 0.7732 R Square 0.5978 Adjusted R Square 0.5476 Standard Error 3.0414 Observations 10 ANOVA df SS MS F Significance F Regression 1 110 110 11.892 0.009 Residual 8 74 9.25 Total 9 184 Coefficients Standard Error t Stat P-value Intercept 39.222 5.942 6.600 0.000 x -0.556 0.161 -3.448 0.009 e. 59.783% of the variability in Y is explained by the variability in X.
The following data represent a company's yearly sales volume and its advertising expenditure over a period of 8 years. (Y) Sales in Millions of Dollars (X) Advertising in ($10,000) 15 32 16 33 18 35 17 34 16 36 19 37 19 39 24 42 a. Develop a scatter diagram of sales versus advertising and explain what it shows regarding the relationship between sales and advertising. b. Use the method of least squares to compute an estimated regression line between sales and advertising. c. If the company's advertising expenditure is $400,000, what are the predicted sales? Give the answer in dollars. d. What does the slope of the estimated regression line indicate? e. Compute the coefficient of determination and fully interpret its meaning. f. Use the F test to determine whether or not the regression model is significant at α = 0.05. g. Use the t test to determine whether the slope of the regression model is significant at α = 0.05. h. Develop a 95% confidence interval for predicting the average sales for the years when $400,000 was spent on advertising. i. Compute the correlation coefficient.
ANS: a. The scatter diagram shows a positive relation between sales and advertising. b. = -10.42 + 0.7895X c. $21,160,000 d. As advertising is increased by $10,000, sales are expected to increase by $789,500. e. 0.8459; 84.59% of variation in sales is explained by variation in advertising f. F = 32.93; p-value (actual p-value using Excel = 0.0012) < .05; reject Ho; it is significant (critical F = 5.99) g. t = 5.74; p-value (actual p-value using Excel = 0.0012) < .05; reject Ho; significant (critical t = 2.447) h. $19,460,000 to $22,860,000 i. 0.9197
Below you are given a partial computer output based on a sample of 8 observations, relating an independent variable (x) and a dependent variable (y). Coefficient Standard Error Intercept -9.462 7.032 x 0.769 0.184 Analysis of Variance SOURCE SS Regression 400 Error (Residual) 138 a. Develop the estimated regression line. b. At α = 0.05, test for the significance of the slope. c. At α = 0.05, perform an F test. d. Determine the coefficient of determination.
ANS: a. = -9.462 + 0.769x b. t = 4.17; p-value (actual p-value using Excel = 0.0059) < .05; reject Ho c. F = 17.39; p-value (actual p-value using Excel = 0.0059) < .05; reject Ho d. 0.743
Assume you have noted the following prices for books and the number of pages that each book contains. Book Pages (x) Price (y) A 500 $7.00 B 700 7.50 C 750 9.00 D 590 6.50 E 540 7.50 F 650 7.00 G 480 4.50 a. Develop a least-squares estimated regression line. b. Compute the coefficient of determination and explain its meaning. c. Compute the correlation coefficient between the price and the number of pages. Test to see if x and y are related. Use α = 0.10.
ANS: a. = 1.0416 + 0.0099x b. r 2 = .5629; the regression equation has accounted for 56.29% of the total sum of squares c. rxy = 0.75 t = 2.54 > 2.015 (df = 5); p-value is between .05 and 0.1; (Excel's results: p-value of 0.052); reject Ho, and conclude x and y are related
A company has recorded data on the daily demand for its product (Y in thousands of units) and the unit price (X in hundreds of dollars). A sample of 15 days demand and associated prices resulted in the following data. ΣX = 75 Σ (Y- )(X- ) = -59 ΣY = 135 Σ (X- )2 = 94 Σ (Y- )2 = 100 SSE = 62.9681 a. Using the above information, develop the least-squares estimated regression line and write the equation. b. Compute the coefficient of determination. c. Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let α = 0.05. d. Would the demand ever reach zero? If yes, at what price would the demand be zero?
ANS: a. = 12.138 - 0.6277X b. R2 = 0.3703 c. F = 7.65; p-value is between .01 and .025; reject Ho and conclude that demand and unit price are related (critical F = 4.67) d. Yes, at $1,934
Below you are given a partial computer output based on a sample of 8 observations, relating an independent variable (x) and a dependent variable (y). Coefficient Standard Error Intercept 13.251 10.77 X 0.803 0.385 Analysis of Variance SOURCE SS Regression Error (Residual) 41.674 Total 71.875 a. Develop the estimated regression line. b. At α = 0.05, test for the significance of the slope. c. At α = 0.05, perform an F test. d. Determine the coefficient of determination.
ANS: a. = 13.251 + 0.803x b. t = 2.086; p-value is between .05 and .1 (critical t = 2.447); do not reject Ho c. F = 4.348; p-value is between .05 and .1 (critical F = 5.99); do not reject Ho d. 0.42
Given below are seven observations collected in a regression study on two variables, X (independent variable) and Y (dependent variable). X Y 2 12 3 9 6 8 7 7 8 6 7 5 9 2 a. Develop the least squares estimated regression equation. b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. c. Perform an F test to determine whether or not the model is significant. Let α = 0.05. d. Compute the coefficient of determination.
ANS: a. = 13.75 -1.125X b. t = -5.196; p-value (actual p-value using Excel = 0.0001) < α = .05; reject Ho (critical t = 2.571) c. F = 27; p-value (actual p-value using Excel = 0.0001) < α = .05; reject Ho (critical F = 6.61) d. 0.844
The following data shows the yearly income (in $1,000) and age of a sample of seven individuals. Income (in $1,000) Age 20 18 24 20 24 23 25 34 26 24 27 27 34 27 a. Develop the least squares regression equation. b. Estimate the yearly income of a 30-year-old individual. c. Compute the coefficient of determination. d. Use a t test to determine whether the slope is significantly different from zero. Let α = 0.05. e. At 95% confidence, perform an F test and determine whether or not the model is significant.
ANS: a. = 16.204 + 0.3848X b. $27,748 c. 0.2266 d. t = 1.21; p-value (actual p-value using Excel = 0.2803) > α = .05; not significant (critical t = 2.571) e. F = 1.46; p-value (actual p-value using Excel = 0.2803) > α = .05; not significant (critical F = 6.61)
Below you are given information on annual income and years of college education. Income (In Thousands) Years of College 28 0 40 3 36 2 28 1 48 4 a. Develop the least squares regression equation. b. Estimate the yearly income of an individual with 6 years of college education. c. Compute the coefficient of determination. d. Use a t test to determine whether the slope is significantly different from zero. Let α = 0.05. e. At 95% confidence, perform an F test and determine whether or not the model is significant.
ANS: a. = 25.6 + 5.2X b. $56,800 c. 0.939 d. t = 6.789; p-value (actual p-value using Excel = 0.0008) < α = .05; reject Ho; significant (critical t = 3.182 e. F = 46.091; p-value (actual p-value using Excel = 0.0008) < α = .05; reject Ho; significant (critical F = 10.13)
The following data represent the number of flash drives sold per day at a local computer shop and their prices. Price (x) Units Sold (y) $34 3 36 4 32 6 35 5 30 9 38 2 40 1 a. Develop a least-squares regression line and explain what the slope of the line indicates. b. Compute the coefficient of determination and comment on the strength of relationship between x and y. c. Compute the sample correlation coefficient between the price and the number of flash drives sold. Use α= 0.01 to test the relationship between x and y.
ANS: a. = 29.7857 - 0.7286x The slope indicates that as the price goes up by $1, the number of units sold goes down by 0.7286 units. b. r 2 = .8556; the regression equation has accounted for 85.56% of the total sum of squares c. rxy = -0.92 t = -5.44 < -4.032 (df = 5); p-value < .01; (Excel's result: p-value = .0028); reject Ho, and conclude x and y are related
Below you are given a partial computer output based on a sample of 21 observations, relating an independent variable (x) and a dependent variable (y). Predictor Coefficient Standard Error Constant 30.139 1.181 X -0.252 0.022 Analysis of Variance SOURCE SS Regression 1,759.481 Error 259.186 a. Develop the estimated regression line. b. At α = 0.05, test for the significance of the slope. c. At α = 0.05, perform an F test. d. Determine the coefficient of determination. e. Determine the coefficient of correlation.
ANS: a. = 30.139 - 0.252X b. t = -11.357; p-value (almost zero) < α = .05; reject Ho (critical t = 2.093) c. F = 128.982; p-value (almost zero) < α = .05; reject Ho (critical F = 4.38) d. 0.872 e. -0.934
Below you are given information on a woman's age and her annual expenditure on purchase of books. Age Annual Expenditure ($) 18 210 22 180 21 220 28 280 a. Develop the least squares regression equation. b. Compute the coefficient of determination. c. Use a t test to determine whether the slope is significantly different from zero. Let α = 0.05. d. At 95% confidence, perform an F test and determine whether or not the model is significant.
ANS: a. = 54.834 + 7.536X b. R2 = 0.568 c. t = 1.621; p-value (actual p-value using Excel = 0.2464) > α = .05; do not reject Ho; not significant (critical t = 4.303) d. F = 2.628; p-value (actual p-value using Excel = 0.2464) > α = .05; do not reject Ho; not significant (critical F = 18.51)
Below you are given a partial computer output based on a sample of 14 observations, relating an independent variable (x) and a dependent variable (y). Predictor Coefficient Standard Error Constant 6.428 1.202 X 0.470 0.035 Analysis of Variance SOURCE SS Regression 958.584 Error (Residual) Total 1021.429 a. Develop the estimated regression line. b. At α = 0.05, test for the significance of the slope. c. At α = 0.05, perform an F test. d. Determine the coefficient of determination. e. Determine the coefficient of correlation.
ANS: a. = 6.428 + 0.47x b. t = 13.529; p-value (actual p-value using Excel = 0.0000) < .05; reject Ho (critical t = 2.179) c. F = 183.04; p-value (actual p-value using Excel = 0.0000) < .05; reject Ho (critical F = 4.75) d. 0.938 e. 0.968
The following data show the results of an aptitude test (Y) and the grade point average of 10 students. Aptitude Test Score (Y) GPA (X) 26 1.8 31 2.3 28 2.6 30 2.4 34 2.8 38 3.0 41 3.4 44 3.2 40 3.6 43 3.8 a. Develop a least squares estimated regression line. b. Compute the coefficient of determination and comment on the strength of the regression relationship. c. Is the slope significant? Use a t test and let α = 0.05. d. At 95% confidence, test to determine if the model is significant (i.e., perform an F test).
ANS: a. = 8.171 + 9.4564X b. 0.83; there is a fairly strong relationship c. t = 6.25; p-value (actual p-value using Excel = 0.0002) < α =.05; it is significant (critical t = 2.306) d. F = 39.07; p-value (actual p-value using Excel = 0.0002) < α =.05; it is significant (critical F = 5.32)
Given below are five observations collected in a regression study on two variables x (independent variable) and y (dependent variable). x y 10 7 20 5 30 4 40 2 50 1 a. Develop the least squares estimated regression equation b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. c. Perform an F test to determine whether or not the model is significant. Let α = 0.05. d. Compute the coefficient of determination. e. Compute the coefficient of correlation.
ANS: a. = 8.3 - 0.15x b. t = -15; p-value (actual p-value using Excel = 0.0001) < .05; reject Ho (critical t = 3.18) c. F = 225; p-value (actual p-value using Excel = 0.0001) < .05; reject Ho (critical F = 10.13) d. 0.9868 e. 0.9934
Shown below is a portion of the computer output for a regression analysis relating sales (Y in millions of dollars) and advertising expenditure (X in thousands of dollars). Predictor Coefficient Standard Error Constant 4.00 0.800 X 0.12 0.045 Analysis of Variance SOURCE DF SS Regression 1 1,400 Error 18 3,600 a. What has been the sample size for the above? b. Perform a t test and determine whether or not advertising and sales are related. Let α = 0.05. c. Compute the coefficient of determination. d. Interpret the meaning of the value of the coefficient of determination that you found in Part c. Be very specific. e. Use the estimated regression equation and predict sales for an advertising expenditure of $4,000. Give your answer in dollars.
ANS: a. 20 b. t = 2.66; p-value is between 0.01 and 0.02; they are related (critical t = 2.101) c. R2 = 0.28 d. 28% of variation in sales is explained by variation in advertising expenditure. e. $4,480,000
Jason believes that the sales of coffee at his coffee shop depend upon the weather. He has taken a sample of 6 days. Below you are given the results of the sample. Cups of Coffee Sold Temperature 350 50 200 60 210 70 100 80 60 90 40 100 a. Which variable is the dependent variable? b. Compute the least squares estimated line. c. Compute the correlation coefficient between temperature and the sales of coffee. d. Is there a significant relationship between the sales of coffee and temperature? Use a .05 level of significance. Be sure to state the null and alternative hypotheses. e. Predict sales of a 90 degree day.
ANS: a. Cups of coffee sold b. = 605.714 - 5.943X c. 0.95197 d. H0: β1 = 0 Ha: β1 0 t = -6.218; p-value (actual p-value using Excel = 0.0034) < α = .05; reject Ho (critical t = 2.776) e. 70.8 or 71 cups
The owner of a retail store randomly selected the following weekly data on profits and advertising cost. Week Advertising Cost ($) Profit ($) 1 0 200 2 50 270 3 250 420 4 150 300 5 125 325 a. Write down the appropriate linear relationship between advertising cost and profits. Which is the dependent variable? Which is the independent variable? b. Calculate the least squares estimated regression line. c. Predict the profits for a week when $200 is spent on advertising. d. At 95% confidence, test to determine if the relationship between advertising costs and profits is statistically significant. e. Calculate the coefficient of determination.
ANS: a. E(Y) = β0 + β1X, where Y is profit and X is advertising cost b. = 210.0676 + 0.80811X c. $371.69 d. t = 6.496; p-value (actual p-value using Excel = 0.0013) < α = .05; reject Ho; relationship is significant (critical t = 3.182) e. 0.9336
The following data represent the number of flash drives sold per day at a local computer shop and their prices. Price (x) Units Sold (y) $34 3 36 4 32 6 35 5 30 9 38 2 40 1 a. Perform an F test and determine if the price and the number of flash drives sold are related. Let α = 0.01. b. Perform a t test and determine if the price and the number of flash drives sold are related. Let α = 0.01.
ANS: a. F = 29.624 > 16.26; p-value < .01; (Excel's result: p-value = .0028); reject Ho, x and y are related b. t = -5.4428 < -4.032; p-value < .01; (Excel's result: p-value = .0028); reject Ho, x and y are related
Assume you have noted the following prices for books and the number of pages that each book contains. Book Pages (x) Price (y) A 500 $7.00 B 700 7.50 C 750 9.00 D 590 6.50 E 540 7.50 F 650 7.00 G 480 4.50 a. Perform an F test and determine if the price and the number of pages of the books are related. Let α = 0.01. b. Perform a t test and determine if the price and the number of pages of the books are related. Let α = 0.01. c. Develop a 90% confidence interval for estimating the average price of books that contain 800 pages. d. Develop a 90% confidence interval to estimate the price of a specific book that has 800 pages.
ANS: a. F = 6.439 < 16.26; p-value is between 0.1 and 0.2 (Excel's result: p-value = .052); do not reject Ho; conclude x and y are not related b. t = 2.5376 < 4.032; p-value is between 0.1 and 0.2. (Excel's result: p-value = .052); do not reject Ho; conclude x and y are not related c. $7.29 to $10.63 (rounded) d. $5.62 to $12.31 (rounded)
Researchers have collected data on the hours of television watched in a day and the age of a person. You are given the data below. Hours of Television Age 1 45 3 30 4 22 3 25 6 5 a. Determine which variable is the dependent variable. b. Compute the least squares estimated line. c. Is there a significant relationship between the two variables? Use a .05 level of significance. Be sure to state the null and alternative hypotheses. d. Compute the coefficient of determination. How would you interpret this value?
ANS: a. Hours of Television b. = 6.564 - 0.1246X c. H0: β1 = 0 Ha: β1 0 t = -12.018; p-value (actual p-value using Excel = 0.0002) < α = .05; reject H0 (critical t = 3.18) d. 0.98 (rounded); 98 % of variation in hours of watching television is explained by variation in age.
The owner of a bakery wants to analyze the relationship between the expenditure of a customer and the customer's income. A sample of 5 customers is taken and the following information was obtained. Y X Expenditure Income (In Thousands) .45 20 10.75 19 5.40 22 7.80 25 5.60 14 The least squares estimated line is = 4.348 + 0.0826 X. a. Obtain a measure of how well the estimated regression line fits the data. b. You want to test to see if there is a significant relationship between expenditure and income at the 5% level of significance. Be sure to state the null and alternative hypotheses. c. Construct a 95% confidence interval estimate for the average expenditure for all customers with an income of $20,000. d. Construct a 95% confidence interval estimate for the expenditure of one customer whose income is $20,000.
ANS: a. R2 = 0.0079 b. H0: β1 = 0 Ha: β1 0 t = 0.154; p-value (actual p-value using Excel = 0.8871) > α = .05; do not reject H0; (critical t = 3.182) c. 0.185 to 12.185 d. -9.151 to 21.151
Given below are five observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). x y 2 4 3 4 4 3 5 2 6 1 a. Develop the least squares estimated regression equation. b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. c. Perform an F test to determine whether or not the model is significant. Let α = 0.05. d. Compute the coefficient of determination. e. Compute the coefficient of correlation.
ANS: Regression Statistics Multiple R 0.970 R Square 0.941 Adjusted R Square 0.922 Standard Error 0.365 Observations 5 ANOVA df SS MS F Significance F Regression 1 6.4 6.400 48.000 0.006 Residual 3 0.4 0.133 Total 4 6.8 Coefficients Standard Error t Stat P-value Intercept 6.000 0.490 12.247 0.001 X -0.800 0.115 -6.928 0.006 a. = 6 - 0.8 x b. p-value < .05; reject Ho c. p-value < .05; reject Ho d. 0.941 e. -0.970
An automobile dealer wants to see if there is a relationship between monthly sales and the interest rate. A random sample of 4 months was taken. The results of the sample are presented below. The estimated least squares regression equation is = 75.061 - 6.254X Y X Monthly Sales Interest Rate (In Percent) 22 9.2 20 7.6 10 10.4 45 5.3 a. Obtain a measure of how well the estimated regression line fits the data. b. You want to test to see if there is a significant relationship between the interest rate and monthly sales at the 1% level of significance. State the null and alternative hypotheses. c. At 99% confidence, test the hypotheses. d. Construct a 99% confidence interval for the average monthly sales for all months with a 10% interest rate. e. Construct a 99% confidence interval for the monthly sales of one month with a 10% interest rate.
ANS: a. R2 = 0.8687 b. H0: β1 = 0 Ha: β1 0 c. test statistic t = -3.64; p-value is between .05 and .10 (critical t = 9.925); do not reject H0 d. -33.151 to 58.199; therefore, 0 to 58.199 e. -67.068 to 92.116; therefore, 0 to 92.116
The following sample data contains the number of years of college and the current annual salary for a random sample of heavy equipment salespeople. Years of College Annual Income (In Thousands) 2 20 2 23 3 25 4 26 3 28 1 29 4 27 3 30 4 33 4 35 a. Which variable is the dependent variable? Which is the independent variable? b. Determine the least squares estimated regression line. c. Predict the annual income of a salesperson with one year of college. d. Test if the relationship between years of college and income is statistically significant at the .05 level of significance. e. Calculate the coefficient of determination. f. Calculate the sample correlation coefficient between income and years of college. Interpret the value you obtain.
ANS: a. Y (dependent variable) is annual income and X (independent variable) is years of college b. = 21.6 + 2X c. $23,600 d. The relationship is not statistically significant since t = 1.51; p-value (actual p-value using Excel = 0.1696) > α = .05 (critical t = 2.306) e. 0.222 f. 0.471; there is a positive correlation between years of college and annual income