Question

Consider the following data sample from the Consumer Reports Restaurant Satisfaction Survey where variable Type indicates whether the restaurant is Italian or a Seafood/steakhouse restaurant. Price indicates average amount paid per person for dinner and drinks. Score reflects diner’s overall satisfaction, with higher values indicating greater satisfaction (100 is completely satisfied). A regression analysis is conducted using several steps to gauge the impact of the explanatory variables on Score (diner’ satisfaction).

Restaurant	Price ($)	Score	Type
Bertucci's	16	77	Italian
Black Angus	24	79	Seafood/Steak
Bonefish Grill	26	85	Seafood/Steak
Bravo!cuccina italiana	18	84	Italian
Buca di Beppo	17	81	Italian
Bugaboo Steak House	18	77	Seafood/Steak
Carrabba's Italian grill	23	86	Italian
Brown's Steakhouse	17	75	Seafood/Steak
Il Fornaio	28	83	Italian
Joe's crab Shack	15	71	Seafood/Steak
Johnny Carino's Italian	17	81	Italian
Lone Star SteakHouse	17	76	Seafood/Steak
Longhorn steakhouse	19	81	Seafood/Steak
Maggio's little Italy	22	83	Italian
McGrath's Fish House	16	81	Seafood/Steak
Oliven Graden	19	79	Italian
Outback Steakhouse	20	82	Italian
Red Lobster	18	81	Seafood/Steak
Romano's macorroni grill	18	82	Italian
The old spaguetti factory	12	79	Italian
Uno Chicago Grill	16	80	Italian

MODEL 2 – Include the dummy variable Dtype which takes value 1 if Italian restaurant, 0 otherwise

1. (1pt) Comment on the goodness of fit of MODEL 2.

   Fully explain here:

2. (3pt) Report the statistical significance of the coefficients for MODEL 2

Fully explain here:

3. (1pt) How important you think the variable Dtype is in explaining Score?

Fully explain here:

4. (0.5pt) Write down the estimated regression equation for this model.

              here:

Answer 1

In order to solve this question I used R software.

R codes and output:

> d=read.table('data.csv',header=T,sep=',')
> head(d)
Price Score Type
1 16 77 1
2 24 79 0
3 26 85 0
4 18 84 1
5 17 81 1
6 18 77 0
> attach(d)
The following objects are masked from d (pos = 3):

Price, Score, Type

> fit=lm(Score~Price+Type)
> summary(fit)

Call:
lm(formula = Score ~ Price + Type)

Residuals:
Min 1Q Median 3Q Max
-5.4202 -2.1048 0.0581 2.4145 4.0592

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 68.6126 3.0505 22.492 1.26e-14 ***
Price 0.5205 0.1546 3.367 0.00344 **
Type 3.0011 1.1661 2.574 0.01913 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.644 on 18 degrees of freedom
Multiple R-squared: 0.4976, Adjusted R-squared: 0.4418
F-statistic: 8.915 on 2 and 18 DF, p-value: 0.002038

> fit2=lm(Score~Price)
> summary(fit2)

Call:
lm(formula = Score ~ Price)

Residuals:
Min 1Q Median 3Q Max
-7.146 -1.875 1.230 1.818 4.301

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 70.3828 3.3833 20.803 1.56e-14 ***
Price 0.5176 0.1760 2.941 0.00839 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.01 on 19 degrees of freedom
Multiple R-squared: 0.3128, Adjusted R-squared: 0.2766
F-statistic: 8.648 on 1 and 19 DF, p-value: 0.008391

a.

F test statistic = 8.915

p-value = 0.0020

Since p-value is less than 0.05, we conclude that model is statistically significant or it fits the given data.

b.

If p-value for each coefficient is less than 0.05, then that variable is statistically significant.

Here we see that both the variables price and type of restaurant have p-value less than 0.05, hence both  variables are statistically significant.

c.

Coefficient of determination for the model without type variable is 31.28% and with type variable is 49.76%. It means adding type variable in regression model will explain 18.48% more variation in the dependent variable score. Hence type of restaurant is an important variable.  

d.

Estimated regression equation is,

Score = 68.6126 + 0.5205 Price + 3.0011 Type