Question

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:

5. (3pt) Interpret the intercept for this model.

  Fully explain here:

6. (1pt) Interpret the coefficient for the variable Dtype.

  Fully explain here:

7. (0.5pt) Curvature from the data is captured by a quadratic and a cubic model using only PRICE. Considering previous MODEL 1, MODEL 2 and the quadratic models below, which one you think fits the data best? Explain why.

Fully explain here:

8. (1pt) Write down the quadratic model equation.

    Fully explain here:

9. (1pt) What is the average Score in the quadratic model?

                  Fully explain here:

Answer 1

Sol:

(a).

R² = 0.498 = 49.8%

Since the model explains less than 55% of the variation, this is not a well-fitted model.

(b).

The hypothesis being tested is:

H₀: β₁ = β₂ = 0

H₁: At least one β_i ≠ 0

The p-value is 0.002.

Since the p-value (0.002) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the slope is significant.

(c).

For every Italian restaurant, the price will increase by 3.0011.

(d) .

The regression equation is:

y = 68.6126 + 0.5205*x1 + 3.0011*x2

R²	0.498
	Adjusted R²	0.442
	R	0.705
	Std. Error	2.644
	n	21
	k	2
	Dep. Var.	Score
ANOVA table
Source	SS	df	MS	F	p-value
Regression	124.6945	2	62.3472	8.92	.0020
Residual	125.8769	18	6.9932
Total	250.5714	20
Regression output				confidence interval
variables	coefficients	std. error	t (df=18)	p-value	95% lower	95% upper
Intercept	68.6126
Price	0.5205	0.1546	3.367	.0034	0.1957	0.8453
Type	3.0011	1.1661	2.574	.0191	0.5512	5.4511

Score	Price	Type
77	16	1
79	24	0
85	26	0
84	18	1
81	17	1
77	18	0
86	23	1
75	17	0
83	28	1
71	15	0
81	17	1
76	17	0
81	19	0
83	22	1
81	16	0
79	19	1
82	20	1
81	18	0
82	18	1
79	12	1
80	16	1

(e).

Keeping the type of restaurant and the price constant, we can expect that the score will be 68.6126 on average.

(f) .

For every Italian restaurant keeping the price constant, we can expect that the score will increase by 3.0011.

(g).

The quadratic model is the best model since it has the highest R-squared value.

(h).

The equation is:

y = 0.0539x² - 7.903x + 305.49

(i).

The average score is when the price is constant, the average score is 305.49.