91Ó°ÊÓ

Zagat restaurant guides publish ratings of restaurants for many large cities around the world (see www.zagat.com). The review for each restaurant gives a verbal summary as well as a 0 - to 30 -point rating of the quality of food, décor, service, and the cost of a dinner with one drink and tip. For 31 French restaurants in Boston in \(2014,\) the food quality ratings had a mean of 24.55 and standard deviation of 2.08 points. The cost of a dinner (in U.S. dollars) had a mean of \(\$ 50.35\) and standard deviation of \(\$ 14.92\). The equation that predicts the cost of a dinner using the rating for the quality of food is \(\hat{y}=-70+4.9 x\). The correlation between these two variables is 0.68 . (Data available in the Zagat_Boston file.) a. Predict the cost of a dinner in a restaurant that gets the (i) lowest observed food quality rating of \(21,\) (ii) highest observed food quality rating of 28 . b. Interpret the slope in context. c. Interpret the correlation. d. Show how the slope can be obtained from the correlation and other information given.

Step by step solution

01

Predict Dinner Cost for Lowest Quality Rating

To predict the dinner cost for a restaurant with the lowest food quality rating, use the given linear equation:\[ \hat{y} = -70 + 4.9x \]Substitute \( x = 21 \) (the lowest observed rating):\[ \hat{y} = -70 + 4.9 \times 21 \]Calculate:\[ \hat{y} = -70 + 102.9 = 32.9 \]Thus, the predicted cost is $32.90.

02

Predict Dinner Cost for Highest Quality Rating

For the highest food quality rating, use the same prediction equation:\[ \hat{y} = -70 + 4.9x \]Substitute \( x = 28 \) (the highest observed rating):\[ \hat{y} = -70 + 4.9 \times 28 \]Calculate:\[ \hat{y} = -70 + 137.2 = 67.2 \]Thus, the predicted cost is $67.20.

03

Interpret the Slope in Context

The slope of the regression line \( 4.9 \) indicates that for each one-point increase in the food quality rating, the average cost of dinner increases by $4.90.

04

Interpret the Correlation

The correlation of 0.68 between the food quality rating and the cost indicates a moderate to strong positive linear relationship. This means that as the quality of food increases, the cost of dinner generally tends to increase as well.

05

Derive the Slope from Correlation and Given Information

The slope \( b \) can also be derived from the formula:\[ b = r \times \left( \frac{s_y}{s_x} \right) \]where \( r = 0.68 \), \( s_y = 14.92 \) (standard deviation of cost), and \( s_x = 2.08 \) (standard deviation of food quality ratings). Substitute these values:\[ b = 0.68 \times \left( \frac{14.92}{2.08} \right) \]Calculate:\[ b = 0.68 \times 7.173 \approx 4.9 \]The calculated slope matches the given slope in the prediction equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient

The correlation coefficient is a statistical measure denoted by the letter \(r\), which quantifies the strength and direction of a relationship between two variables. In this case, the correlation between food quality ratings and dinner costs for the Boston French restaurants is 0.68.
This value tells us several things:

• A correlation of 0.68 is considered moderately strong, meaning there is a clear relationship, but it's not perfect.
• The positive sign indicates that as one variable increases, the other tends to increase as well. For these restaurants, as the quality of food ratings go up, so do the costs of dinner.
• Correlations range from -1 to +1. A correlation close to 1 means a strong positive relationship, while close to -1 means a strong negative relationship. A correlation of 0 implies no linear relationship.

This information helps us predict and understand how changes in one variable (food quality) could affect another (dinner cost). Understanding this correlation improves our confidence in the linear regression predictions.

Prediction Equation

The prediction equation used in this exercise is \( \hat{y} = -70 + 4.9x \). This equation allows us to estimate the cost of dinner based on a given food quality rating.

• \( \hat{y} \) is the predicted cost of dinner.
• \( x \) is the food quality rating.
• -70 is the y-intercept, which represents the starting value of the cost when the food quality rating is zero.
• 4.9 is the slope of the equation, indicating how much the cost changes for each unit increase in the food quality rating.

To make predictions, you substitute the restaurant's food quality rating into the \( x \) value and solve for \( \hat{y} \).
This equation guides us to make educated estimates about dinner costs, ensuring we understand the role of food quality in pricing.

Slope Interpretation

The slope in a linear equation like \( \hat{y} = -70 + 4.9x \) tells us how changes in the independent variable impact the dependent variable.
In this equation:

• The slope is 4.9, which means that for every additional point increase in a restaurant's food quality rating, the predicted cost of dinner increases by $4.90.
• This value helps quantify the relationship between ratings and costs, suggesting that improvements in food quality have a substantial impact on what restaurants can charge.

Understanding slope enables us to see not just that costs increase with quality, but also how significant the increase is, allowing restaurateurs and patrons alike to quantify the value of improved service.

Standard Deviation

Standard deviation is a measure of how spread out numbers are in a dataset. In the context of this exercise, two standard deviations are particularly relevant:

• For food quality ratings: The standard deviation is 2.08. This number shows how much the ratings for food quality vary from the average rating of 24.55 for the restaurants sampled.
• For dinner costs: The standard deviation is \(14.92. It indicates how much the actual dinner costs differ from the average cost of \)50.35.

A lower standard deviation means the data points are closer to the mean, and a higher one indicates more spread. In regression analysis, standard deviation helps us understand the variability of data. It is critical in calculating other statistics, such as the slope of the regression line using the formula \( b = r \times \left( \frac{s_y}{s_x} \right) \), as it measures the "typical" deviation from the average.