91Ó°ÊÓ

Rating restaurants 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

Predicting Cost for Lowest Rating

To find the predicted cost for the lowest observed food quality rating of 21, substitute \( x = 21 \) into the prediction equation: \( \hat{y} = -70 + 4.9 \times 21 \). This calculates to \( \hat{y} = -70 + 102.9 = 32.9 \). Thus, the predicted cost is $32.90.

02

Predicting Cost for Highest Rating

For the highest observed food quality rating of 28, substitute \( x = 28 \) into the prediction equation: \( \hat{y} = -70 + 4.9 \times 28 \). This calculates to \( \hat{y} = -70 + 137.2 = 67.2 \). Thus, the predicted cost is $67.20.

03

Interpreting the Slope

The slope of 4.9 in the context of this problem indicates that for every 1-point increase in the food quality rating, the predicted cost of a dinner increases by $4.90.

04

Interpreting the Correlation

A correlation of 0.68 suggests a moderate to strong positive linear association between the food quality ratings and the cost of dinner. This means that as the quality of food increases, the cost tends to increase as well.

05

Calculating Slope from Correlation

The slope \( b \) of the regression line can be calculated using the formula \( b = r \times \frac{s_y}{s_x} \), where \( r = 0.68 \), \( s_y = 14.92 \) (standard deviation of cost), and \( s_x = 2.08 \) (standard deviation of food quality). Thus, \( b = 0.68 \times \frac{14.92}{2.08} \approx 4.9 \), which matches the given slope of the regression equation.

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Key Concepts

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

Regression Analysis in Predictive Modeling

Regression analysis is a statistical method for examining the relationship between two quantitative variables. In this exercise, we use regression to predict the cost of a dinner at a restaurant based on its food quality rating. The prediction equation provided is \( \hat{y} = -70 + 4.9x \), where \( \hat{y} \) is the predicted cost, and \( x \) is the food quality rating.The goal of regression analysis here is to find how changes in food quality ratings impact the cost of a dinner. The interpretation of the regression's slope, \( 4.9 \), tells us that for every one-point increase in the food quality rating, the predicted dinner cost increases by \( \$4.90 \). This information can be valuable for restaurant goers wishing to balance their budget with the quality of dining experience they seek. Besides prediction, regression helps in understanding the underlying relationship, as shown through the slope calculation and the intercept, which enlights us on the average cost when the food quality rating would theoretically be zero—though this doesn't practically happen here.

Understanding Correlation in Data Analysis

Correlation is a statistical measure that describes the strength and direction of a linear relationship between two variables. In the given scenario, the correlation coefficient between food quality ratings and cost of dinner is 0.68. This value implies a moderate to strong positive relationship. A positive correlation means that as one variable increases, the other variable tends to also increase. Since the correlation coefficient ranges between -1 and 1, a value of 0.68 indicates that a higher food quality rating is likely associated with a higher cost of dinner, and vice versa. This correlation value helps understand that while the relationship is not perfectly linear, there is a general trend that people pay more for better-rated food quality in restaurants. However, it’s important to remember that correlation does not imply causation, and other factors might also influence the cost.

Statistical Interpretation of Regression Components

When interpreting statistical models like regression, it's essential to understand both the slope and intercept. In our exercise, the slope is \( 4.9 \) and the intercept is \(-70\). Here's how these components come together:

• Slope: This indicates the rate of change. For each one-unit increase in the food quality rating, the dinner cost increases by \( \$4.90 \). This quantifies the sensitivity of dinner cost to changes in food quality ratings.
• Intercept: The intercept value \(-70\) represents the predicted cost when the food quality rating is zero. While a zero rating is not practical, the intercept calculation offers a starting point for the regression line on a graph.

Interpreting these components helps us understand how predictive models provide insights into cost estimation based on measurable qualities like food rating.

Understanding Standard Deviation in Context

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It provides insight into the spread of data points around the mean. In our scenario, the standard deviation for food quality ratings is 2.08, while for dinner cost, it is \( \\(14.92 \).These values offer significant insights:

• A standard deviation of 2.08 for food quality ratings suggests that the ratings are relatively close to the average (mean), indicating consistency among the restaurants' food quality.
• The \( \\)14.92 \) standard deviation for the cost indicates that dinner prices have more variability. Cost can fluctuate significantly from the average \( \$50.35 \).

Understanding standard deviation helps assess predictability. Lower variance in food ratings hints at reliability, while higher cost variance suggests that while quality ratings provide insight, the actual dining cost might still significantly fluctuate, impacted by other factors too.