Zagats publishes restaurant ratings for various locations in the United States
Zagats publishes restaurant ratings for various locations in the United States. The file Restaurants (posted in Blackboard) contains the Zagat rating for food, dcor, service, and the cost per person for a sample of 100 restaurants located in the center of New York City and in an outlying area of New York City. Develop a regression model to predict the cost per person, based on a variable that represents the sum of ratings for food dcor, and service.
Construct a Scatter Plot
Assuming a linear relationship, use the least-squares method to compute the regression coefficients b0 and b1.
Interpret the meaning of the Y intercept, b0 and the slope, b1 in this problem.
Predict the mean cost per person for a restaurant with a summated rating of 35.
What should you tell the owner of a group of restaurants in this geographical area about the relationship between the summated rating and the cost of a meal?
The formulas to calculate these coefficients are:
b1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)2
b0 = ȳ - b1 * x̄
Where:
- xi is the summated rating for the i-th restaurant
- yi is the cost per person for the i-th restaurant
- x̄ is the mean of the summated ratings
- ȳ is the mean of the cost per person
- Y-intercept (b0): The y-intercept represents the predicted cost per person when the summated rating is zero. In this context, it might not have a practical meaning, as a restaurant is unlikely to have ratings of zero for food, decor, and service. However, statistically, it's a component of the regression equation.
- Slope (b1): The slope represents the change in the cost per person for each one-unit increase in the summated rating. For example, if b1 is 2.5, this means that, on average, the cost per person increases by $2.50 for every one-point increase in the summated rating.
- ŷ is the predicted cost per person
- b0 is the y-intercept
- b1 is the slope
- x is the summated rating (35 in this case)
- If the slope (b1) is positive, there is a positive relationship between the summated rating and the cost per person. This means that restaurants with higher ratings for food, decor, and service tend to have a higher cost per person.
- The magnitude of the slope (b1) indicates the strength of this relationship. A larger b1 suggests a stronger impact of the summated rating on the cost per person.
- The owner can use this information to understand how their restaurants' ratings influence their pricing. If their restaurants have high ratings, they may justify higher prices.
- However, other factors also influence the cost of a meal, such as location, competition, and the overall economy. The owner should consider these factors in addition to the summated rating when making pricing decisions.
. Construct a Scatter Plot
To visualize the relationship between the summated rating (food, decor, and service) and the cost per person, I would create a scatter plot. The summated rating would be plotted on the x-axis, and the cost per person would be plotted on the y-axis. This plot will help in understanding if a linear relationship exists between the two variables.
2. Compute Regression Coefficients
Assuming a linear relationship, I would use the least-squares method to calculate the regression coefficients b0 (y-intercept) and b1 (slope). The least-squares method minimizes the sum of the squared differences between the observed and predicted values of the cost per person.