Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.
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Fixed Percentage If a restaurant were to change its tipping policy so that a constant tip of 20% of the bill is added to the cost of the dinner, what would be the value of the linear correlation coefficient for the paired amounts of dinners/tips?

검증된 단계별 안내

Understand the problem: We need to find the linear correlation coefficient for a fixed percentage tip policy, where the tip is 20% of the dinner cost.

Recognize that when the tip is a fixed percentage of the dinner cost, the relationship between dinner cost and tip is perfectly linear. This means that the tip is directly proportional to the dinner cost.

Recall that the linear correlation coefficient, denoted as \( r \), measures the strength and direction of a linear relationship between two variables. When one variable is a constant multiple of another, the correlation coefficient is either +1 or -1, depending on the direction of the relationship.

Since the tip is 20% of the dinner cost, the relationship is positive and direct. Therefore, the correlation coefficient \( r \) will be +1, indicating a perfect positive linear relationship.

Conclude that the value of the linear correlation coefficient for the paired amounts of dinners/tips, under a fixed 20% tipping policy, is +1.

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주요 개념

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Linear Correlation Coefficient

The linear correlation coefficient, often denoted as 'r', measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 or -1 indicate a strong linear relationship, and values near 0 suggest a weak linear relationship. In this context, it helps determine how well the tips correlate with the dinner costs.

Fixed Percentage Tipping

Fixed percentage tipping involves adding a constant percentage of the bill as a tip, regardless of the bill's total amount. In this scenario, a 20% tip is added to each dinner cost, creating a direct proportional relationship between the dinner cost and the tip amount. This proportionality implies that the correlation coefficient would be perfect, as the tip is a fixed percentage of the dinner cost.

Proportional Relationships

A proportional relationship is one where two quantities increase or decrease at the same rate, maintaining a constant ratio. In the case of a fixed percentage tip, the tip amount is always a constant fraction of the dinner cost, leading to a perfect linear relationship. This means the correlation coefficient would be exactly 1, indicating a perfect positive linear relationship.