Zagat Restaurant Ratings Analysis In New York City And Surrounding Areas

Simple Linear Regression Analysis of Restaurant Ratings and Cost Per Meal

Question:

Write an essay on restaurant ratings for various locations in the United States.

The report discusses about the Zagat’s publishes restaurant ratings for various locations in the United States. The file Restaurants contains the Zagat rating for food, décor, service and the cost per person for a sample of 100 restaurants located in New York City and in a suburb of New York. The analysis on the data sets is done to determine the summated ratings and the costs of restaurant meals in the New York City and its surrounded areas. The report briefly explains the restaurant industry, its cost structure, its relationship with customers and their satisfactions, the quality of meals provided and moreover the correlation between costs of restaurant meal and services with the summated restaurant ratings. The assignment is done through simple linear regression for determining the correlation among the chosen variables. The New York City is full of these types of restaurants. Due to the city’s huge diversity and spread, the restaurants are one of the biggest industries in that particular region. As there is diversified population staying and wandering, many tastes and types of restaurant meals are available in the NYC. Therefore, this report will broadly show with the data analysis that how the summated ratings of those restaurants are important in relation with the restaurant cost of meal.

Zagat’s publishes restaurant ratings for various locations in the United States. The file Restaurants contains the Zagat rating for food, décor, service and the cost per person for a sample of 100 restaurants located in New York City and in a suburb of New York.  On the given data, a quantitative data analysis is done by simple linear regression taking the two variables as summated ratings and the cost of meals in the restaurants. Regression is the technique that interprets the relationship between two or more variables (Montgomery, Douglas, Elizabeth Peck, and Geoffrey 2015). Here the simple linear regression technique is applied to interpret the relationship between the two variables.Firstly, to show the relationship between the summated rating and cost of meals in the restaurant, the scatter plot is done by taking the dependent variable to be the summated ratings and independent variable being the cost of meals.

Table 1: Plate gap and tear:

Plate Gap	Tear
0.00	0.00
0.00	0.00
1.80	0.45
1.80	0.85
0.00	0.35
0.00	0.30
0.00	0.70
0.00	1.90
0.00	0.25
-1.80	0.10
-1.80	0.15
3.00	3.90
-1.80	0.00
0.00	0.55
-3.00	0.00
-1.80	0.05
1.80	0.40
1.80	4.30
0.00	0.00

The above scatter plot shows a linear and positive trend line which is upward sloping. This means the relationship between the two variables is positive and increasing i.e. the high the value of number of plates, the high will be the rating in terms of the reviews. The rise in plates means rise in production and rise in revenue by the restaurant too. Therefore, this will cause higher ratings.

The regression output is found through simple linear regression analysis that is given below:

Table 2: Simple Linear Regression Output:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.617347283	(correlation coefficient – strenght and direction of relationship (exel does not give us the sign) – from scatterplot we know it is positive
R Square	0.381117667	(coefficient of determination)
Adjusted R Square	0.344712824
Standard Error	1.024120857
Observations	19	(
ANOVA
	df	SS	MS	F	Significance F
Regression	1	10.98	10.98	10.46887269	0.004860746
Residual	17	17.83	1.048823529
Total	18	28.81
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept (b0)	0.75	0.234949438	3.192176183	0.00533663	0.254300016	1.2457	0.2543	1.2457
Plate Gap (b1)	0.5	0.154532576	3.235563736	0.004860746	0.173964763	0.826035	0.173965	0.826035

 Now, it is assumed and shown that the two variables are having a linear cost relationship. Therefore, the values of the coefficients b₀ and b₁ are- 0.75 and 0.5 respectively.

According to the question when plate gap (x) increases by 1 unit, the predicted value of tear rating (y) increases by 0.5 units. Therefore, b₀ and b₁ have relationship which is greater than zero i.e. positive. A rise in plate gap of 1 unit will cause the tear rating increase of 0.5 units. At the same time, the positive intercept means that the curve starts from a positive value with positive slope.

Summary Output

Predicted Y = 0.75 + 0.5x

Predicted Tear rating = 0.75 + 0.5 (plate gap)

When plate gap is 0, y = mx + c –> 0.75 + 0.5(0) = 0.75

It means that when plate gap (x) is zero, it means we are looking at the intercept, b0.

Predicted tear rating = 0.75 + 0.5 (0) = 0.75

Therefore, the estimated simple linear regression equation is given by:

Predicted Y = 0.75 + 0.5x

Now, when the rating is 50, the predicted value of cost per plate is given by:

50= 0.75 + 0.5 (cost per plate)

è 50-0.75= 0.5x

• 5 x = 49.25
• x= 49.25/0.5
• x= 98.50

Here the residual analysis is done with the data on size of the restaurant and the value of rent. The data is given in the table below:

Size	Rent
850	950
1450	1600
1085	1200
1232	1500
718	950
1485	1700
1136	1650
726	935
700	875
956	1150
1100	1400
1285	1650
1985	2300
1369	1800
1175	1400
1225	1450
1245	1100
1259	1700
1150	1200
896	1150
1361	1600
1040	1650
755	1200
1000	800
1200	1750

Table 3: Size and rent data on restaurants:

The above data shows a linear relationship with a positive trend. This means the size effect has a rising trend with the rent of a restaurant. Positive relationship indicates an increasing trend which means as size of a restaurant rises, the value of rent also rises. This has direct relationship with cost as rent is the fixed cost included in the total cost factor. Therefore, as size of the restaurant increases, the rent increases thereby increasing the total cost. Thus, the ratings will be affected with this cost value. If the cost is not overcome with the revenue and if the number of plates served is not high, the ratings will fall. This means rise in cost may cause fall in the total profit as well as the summated ratings regarding the restaurant meals.

The summary output of regression analysis is given as below:

Table 4: Summary output of regression between size of the restaurant and rent

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.850060796
R Square	0.722603356
Adjusted R Square	0.710542633
Standard Error	194.5953946
Observations	25
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2268777	2268777	59.91376	7.52E-08
Residual	23	870949.5	37867.37
Total	24	3139726
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept (b0)	177.1208202	161.0043	1.1001	0.28267	-155.942	510.1835	-155.942	510.1835
Size (b1)	1.065143906	0.137608	7.740398	7.52E-08	0.780479	1.349809	0.780479	1.349809

The above output indicates a predicted value for the size and rent equation, which is:

From output, can see that b₀ = 177.12, and b₁ = 1.0651(x)

Predicted rent = 117.12 + 1.0651 (Size)

When the size of the apartment increases by 1 square feet, the predicted rent increases by $1.065

Y = mx + c –> Y = 177.1208 + 1.0651 (1000) = $1242.12

Predicted rent = 177.12 + 1.0651 (1000) = $1242.12

The histogram is formed regarding the values:

Bin range	Frequency	Cumulative %
500	0	0.00%
1000	13	26.00%
1500	26	78.00%
2000	10	98.00%
More	1	100.00%

 The cumulative frequency shows a rising manner. This rising frequency is shown in the linear trend equation. Positive relationship indicates an increasing trend which means as size of a restaurant rises, the value of rent also rises. This has direct relationship with cost as rent is the fixed cost included in the total cost factor (Cameron, Colin and Trivedi 2013). Therefore, as size of the restaurant increases, the rent increases thereby increasing the total cost. Thus, the ratings will be affected with this cost value. If the cost is not overcome with the revenue and if the number of plates served is not high, the ratings will fall. This means rise in cost may cause fall in the total profit as well as the summated ratings regarding the restaurant meals. The data is shown through the histogram presentation above.

The above diagram is the representation of asking rest with the predicted rent of the restaurant. The values show rising manner in the data above.

Size and Rent Data on Restaurants

Now, taking all the outputs, summarized results can be found from the regression analysis. The regression results for size effects with the rent are given below:

Regression Statistics
Multiple R	0.850060796
R Square	0.722603356
Adjusted R Square	0.710542633
Standard Error	194.5953946
Observations	25

 The above value of multiple R is 0.850060796, near to 1. Generally, the value of correlation coefficient i.e. R lies between -1 to 1. At the same time, the value of R square is 0.722603356. However, the value of R in this case is high, near to 1, meaning that there is a high correlation between the results for size effects with the rent (Chatterjee, Samprit and Hadi 2015). This has direct relationship with cost as rent is the fixed cost included in the total cost factor. Therefore, as size of the restaurant increases, the rent increases thereby increasing the total cost. Thus, the ratings will be affected with this cost value. If the cost is not overcome with the revenue and if the number of plates served is not high, the ratings will fall. This means rise in cost may cause fall in the total profit as well as the summated ratings regarding the restaurant meals.

ANOVA
	df	SS	MS	F	Significance F
Regression	1	2268776.545	2268777	59.91376	7.52E-08
Residual	23	870949.4547	37867.37
Total	24	3139726

 In the ANOVA output, the value of F statistic is also significant. Therefore, the variances are compared between the two variables i.e. size of the restaurant and rent, has a low variability. This also indicates a direct relation between them.

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept (b0)	177.1208202	161.0042766	1.1001	0.28267	-155.942	510.1835	-155.942	510.1835
Size (b1)	1.065143906	0.137608412	7.740398	7.52E-08	0.780479	1.349809	0.780479	1.349809

 The values of intercept are signified by the above results. The value of standard error is high. The t statistic is higher than 1, meaning a highly significant relationship between the variables (Draper, Norman and Harry Smith 2014). The p-value is low. The low value of p signifies the model to be more significant. Therefore, the overall data structure indicates a good fitted model on which the cost should be reduced in order to make profit in the restaurants as size of the restaurant directly affects the rent and thereby the fixed cost.

Conclusion:

The report has dealt with the Zagat’s publishes restaurant ratings for various locations in the United States. The file Restaurants contains the Zagat rating for food, décor, service and the cost per person for a sample of 100 restaurants located in New York City and in a suburb of New York. The analysis on the data sets is done to determine the summated ratings and the costs of restaurant meals in the New York City and its surrounded areas. All the above data has indicated a positive relationship with the summated ratings in terms of cost of the restaurant. To eradicate or reduce the cost, the size factor should be taken into account, as the rise in size causes a rise in the cost. Moreover, the relationship between the plates served and the ratings are calculated which gives a clear picture that as there is more plates served, the ratings are high. As there is diversified population staying and wandering, many tastes and types of restaurant meals are available in the NYC. Therefore, this report shows broadly with the data analysis that how the summated ratings of those restaurants are important in relation with the restaurant cost of meal.

References:

Cameron, A. Colin, and Pravin K. Trivedi. Regression analysis of count data. Vol. 53. Cambridge university press, 2013.

Chatterjee, Samprit, and Ali S. Hadi. Regression analysis by example. John Wiley & Sons, 2015.

Draper, Norman R., and Harry Smith. Applied regression analysis. John Wiley & Sons, 2014.

Montgomery, Douglas C., Elizabeth A. Peck, and G. Geoffrey Vining.Introduction to linear regression analysis. John Wiley & Sons, 2015.