For each of 10 popular prescription drugs, file XR15042 lists the retail price (in U.S. dollars) for the drug in several different countries, including the United States, Canada, Great Britain, and Australia. Determine and interpret the coefficients of correlation and determination for U.S. prices versus Canadian prices. SOURCE: “Prescription Drugs Cheaper in Other Nations,” USA Today, November 10, 1999, p. 2A

XR15042
Drug US Price Canada Price Great Britain Price Australia Price
Prilosec 3.31 1.47 1.67 1.29
Prozac 2.27 1.07 1.08 0.82
Lipitor 2.54 1.34 1.67 1.32
Prevacid 3.13 1.34 0.82 0.83
Epogen 23.40 21.44 27.48 29.24
Zocor 3.16 1.47 1.73 1.75
Zoloft 1.98 1.07 0.95 0.84
Zyprexa 5.27 3.39 2.86 2.63
Claritin 1.96 1.11 0.41 0.48
Paxil 2.22 1.13 1.70 0.82

To determine the coefficients of correlation and determination for U.S. prices versus Canadian prices, we will first calculate the correlation coefficient and then use it to find the coefficient of determination. The correlation coefficient measures the strength and direction of the linear relationship between two variables, while the coefficient of determination represents the proportion of the total variation in one variable that can be explained by the other variable.

Here are the steps to calculate the correlation coefficient:

Step 1: Create a table to organize the data for U.S. prices and Canadian prices:

| Drug | US Price | Canada Price |
|-------------|----------|--------------|
| Prilosec | 3.31 | 1.47 |
| Prozac | 2.27 | 1.07 |
| Lipitor | 2.54 | 1.34 |
| Prevacid | 3.13 | 1.34 |
| Epogen | 23.40 | 21.44 |
| Zocor | 3.16 | 1.47 |
| Zoloft | 1.98 | 1.07 |
| Zyprexa | 5.27 | 3.39 |
| Claritin | 1.96 | 1.11 |
| Paxil | 2.22 | 1.13 |

Step 2: Calculate the mean (average) of the US prices and the Canadian prices:

Mean of US prices = (3.31 + 2.27 + 2.54 + 3.13 + 23.40 + 3.16 + 1.98 + 5.27 + 1.96 + 2.22) / 10 = 4.745

Mean of Canadian prices = (1.47 + 1.07 + 1.34 + 1.34 + 21.44 + 1.47 + 1.07 + 3.39 + 1.11 + 1.13) / 10 = 4.469

Step 3: Calculate the deviations of the US prices and the Canadian prices from their means:

Deviation of US prices = US Price - Mean of US prices
Deviation of Canadian prices = Canada Price - Mean of Canadian prices

Step 4: Calculate the product of the deviations of the US prices and the Canadian prices:

Product of deviations = Deviation of US prices * Deviation of Canadian prices

Step 5: Calculate the sums of the products of deviations, the sums of the squared deviations, and the square roots of the squared deviations for both the US prices and the Canadian prices:

Sum of products of deviations = (Product of deviations1 + Product of deviations2 + ... + Product of deviations10)

Sum of squared deviations of US prices = ((Deviation of US prices1)^2 + (Deviation of US prices2)^2 + ... + (Deviation of US prices10)^2)

Sum of squared deviations of Canadian prices = ((Deviation of Canadian prices1)^2 + (Deviation of Canadian prices2)^2 + ... + (Deviation of Canadian prices10)^2)

Step 6: Calculate the correlation coefficient using the formulas:

Correlation coefficient = Sum of products of deviations / (Square root of (Sum of squared deviations of US prices * Sum of squared deviations of Canadian prices))

Step 7: Interpret the correlation coefficient:

The correlation coefficient will range from -1 to 1. If the correlation coefficient is close to 1, it indicates a strong positive linear relationship, meaning that as US prices increase, Canadian prices also tend to increase. If the correlation coefficient is close to -1, it indicates a strong negative linear relationship, meaning that as US prices increase, Canadian prices tend to decrease. If the correlation coefficient is close to 0, it indicates a weak or no linear relationship between the two variables.

To find the coefficient of determination, we need to square the correlation coefficient. The coefficient of determination represents the proportion of the total variation in one variable that can be explained by the other variable.