FIGURE 12.20 Minitab Regression Analysis of the Airline Cost Example Although the Excel regression output, shown in

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FIGURE 12.20 Minitab Regression Analysis of the Airline Cost Example Although the Excel regression output, shown in Figure 12.21 for Demonstration Problem 12.1, is somewhat different from the Minitab output, the same essential regression features are present. The regression equation is found under Coefficients at the bottom of ANOVA. The slope or coefficient of x is 2.2315 and the y-intercept is 30.9125. The standard error of the estimate for the hospital problem is given as the fourth statistic under Regression Statistics at the top of the output, Standard Error = 15.6491. The r2 value is given as 0.886 on the second line. The t test for the slope is found under t Stat near the bottom of the ANOVA section on the “Number of Beds” (x variable) row, t = 8.83. Adjacent to the t Stat is the p-value, which is the probability of the t statistic occurring by chance if the null hypothesis is true. For this slope, the probability shown is 0.000005. The ANOVA table is in the middle of the output with the F value having the same probability as the t statistic, 0.000005, and equaling t2. The predicted values and the residuals are shown in the Residual Output section.

FIGURE 12.21 Excel Regression Output for Demonstration Problem 12.1 Decision Dilemma Solved Predicting International Hourly Wages by the Price of a Big Mac In the Decision Dilemma, questions were raised about the relationship between the price of a Big Mac hamburger and net hourly wages around the world and if a model could be developed to predict net hourly wages by the price of a Big Mac. Data were given for a sample of 20 countries. In exploring the possibility that there is a relationship between these two variables, a Pearson product-moment correlation coefficient, r, was computed to be .761. This r value indicates that there is a relatively high correlation between the two variables and that developing a regression model to predict one variable by the other has potential. Designating net hourly wages as the y or dependent variable and the price of a Big Mac as the x or predictor variable, the following regression output was obtained for these data using Excel.

SUMMARY OUTPUT
Regression Statistics
Multiple R		0.761
R Square		0.579
Adjusted R Square		0.555
Standard Error		4.2145
Observations		20
ANOVA
	df	SS	MS	F	Significance F
Regression	1	438.85	438.85	24.71	0.0001
Residual	18	319.71	17.76
Total	19	758.56

		Coefficients	Standard Error	t Stat	P-value
Intercept		−1.431	4.114	−0.35	0.7319
Big Mac Price (U.S.$)		4.766	0.959	4.97	0.0001

Taken from this output, the regression model is

Net Hourly Wage = −1.431 + 4.766 (Price of Big Mac)

While the y-intercept has virtually no practical meaning in this analysis, the slope indicates that for every dollar increase in the price of a Big Mac, there is an incremental increase of $4.766 in net hourly wages for a country. It is worth underscoring here that just because there is a relationship between two variables, it does not mean there is a cause-and-effect relationship. That is, McDonald’s cannot raise net hourly wages in a country just by increasing the cost of a Big Mac!

Using this regression model, the net hourly wage for a country with a $3.00 Big Mac can be predicted by substituting x = 3 into the model.

Net Hourly Wage = −1.431 + 4.766(3) = $12.87

That is, the model predicts that the net hourly wage for a country is $12.87 when the price of a Big Mac is $3.00.

How good a fit is the regression model to the data? Observe from the Excel output that the F value for testing the overall significance of the model (24.71) is highly significant with a p-value of .0001, and that the t statistic for testing to determine if the slope is significantly different from zero is 4.97 with a p-value of .0001. In simple regression, the t statistic is the square root of the F value and these statistics relate essentially the same information—that there are significant regression effects in the model. The r² value is 57.9%, indicating that the model has moderate predictability. The standard error of the model, s = 4.2145, indicates that if the error terms are approximately normally distributed, about 68% of the predicted net hourly wages would fall within ±$4.2145.

Shown here is an Excel-produced line fit plot. Note from the plot that there generally appears be a linear relationship between the variables but that many of the data points fall considerably away from the fitted regression line, indicating that the price of a Big Mac only partially accounts for net hourly wages.

Decision Dilemma Solved

Predicting International Hourly Wages by the Price of a Big Mac

In the Decision Dilemma, questions were raised about the relationship between the price of a Big Mac hamburger and net hourly wages around the world and if a model could be developed to predict net hourly wages by the price of a Big Mac. Data were given for a sample of 20 countries. In exploring the possibility that there is a relationship between these two variables, a Pearson product-moment correlation coefficient, r, was computed to be .761. This r-value indicates that there is a relatively high correlation between the two variables and that developing a regression model to predict one variable by the other has potential. Designating net hourly wages as the y or dependent variable and the price of a Big Mac as the x or predictor variable, the following regression output was obtained for these data using Excel.