Question

The Consumer Reports Restaurant Customer Satisfaction Survey is based upon 148,599 visits to full-service restaurant chains.† Assume the following data are representative of the results reported. The variable type indicates whether the restaurant is an Italian restaurant or a seafood/steakhouse. Price indicates the average amount paid per person for dinner and drinks, minus the tip. Score reflects diners' overall satisfaction, with higher values indicating greater overall satisfaction. A score of 80 can be interpreted as very satisfied. (Let x₁ represent average meal price, x₂ represent type of restaurant, and y represent overall customer satisfaction.)

Restaurant	Type	Price ($)	Score
Bertucci's	Italian	16	77
Black Angus Steakhouse	Seafood/Steakhouse	24	79
Bonefish Grill	Seafood/Steakhouse	26	85
Bravo! Cucina Italiana	Italian	18	84
Buca di Beppo	Italian	17	81
Bugaboo Creek Steak House	Seafood/Steakhouse	18	77
Carrabba's Italian Grill	Italian	23	86
Charlie Brown's Steakhouse	Seafood/Steakhouse	17	75
Il Fornaio	Italian	28	83
Joe's Crab Shack	Seafood/Steakhouse	15	71
Johnny Carino's	Italian	17	81
Lone Star Steakhouse & Saloon	Seafood/Steakhouse	17	76
Longhorn Steakhouse	Seafood/Steakhouse	19	81
Maggiano's Little Italy	Italian	22	83
McGrath's Fish House	Seafood/Steakhouse	16	81
Olive Garden	Italian	19	81
Outback Steakhouse	Seafood/Steakhouse	20	80
Red Lobster	Seafood/Steakhouse	18	78
Romano's Macaroni Grill	Italian	18	82
The Old Spaghetti Factory	Italian	12	79
Uno Chicago Grill	Italian	16	76

(a)

Develop the estimated regression equation to show how overall customer satisfaction is related to the independent variable average meal price. (Round your numerical values to two decimal places.)

ŷ =

(b)

At the 0.05 level of significance, test whether the estimated regression equation developed in part (a) indicates a significant relationship between overall customer satisfaction and average meal price. (Use an F test.)

State the null and alternative hypotheses.

H₀: β₁ ≤ 0
H_a: β₁ > 0

H₀: β₁ = 0
H_a: β₁ ≠ 0

    H₀: β₁ = 0
H_a: β₁ > 0

H₀: β₁ ≥ 0
H_a: β₁ < 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Reject H₀. There is sufficient evidence to conclude that there is a significant relationship.

Reject H₀. There is insufficient evidence to conclude that there is a significant relationship.    

Do not reject H₀. There is sufficient evidence to conclude that there is a significant relationship.

Do not reject H₀. There is insufficient evidence to conclude that there is a significant relationship.

(c)

Develop a dummy variable that will account for the type of restaurant (Italian or seafood/steakhouse).

Let x₂ = 0 if the restaurant is an Italian restaurant and x₂ = 0 if the restaurant is a seafood/steakhouse.

Let x₂ = 1 if the restaurant is an Italian restaurant and x₂ = 1 if the restaurant is a seafood/steakhouse.    

Let x₂ = 1 if the restaurant is an Italian restaurant and x₂ = 0 if the restaurant is a seafood/steakhouse.

(d)

Develop the estimated regression equation to show how overall customer satisfaction is related to the average meal price and the type of restaurant. (Use the dummy variable developed in part (c). Round your numerical values to two decimal places.)

ŷ =

(e)

Is type of restaurant a significant factor in overall customer satisfaction? (Use α = 0.05.)

State the null and alternative hypotheses.

H₀: β₂ = 0
H_a: β₂ ≠ 0

H₀: β₂ = 0
H_a: β₂ > 0    

H₀: β₂ ≤ 0
H_a: β₂ > 0

H₀: β₂ ≥ 0
H_a: β₂ < 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Reject H₀. There is insufficient evidence to conclude that the type of restaurant is a significant factor.

Reject H₀. There is sufficient evidence to conclude that the type of restaurant is a significant factor.    

Do not reject H₀. There is insufficient evidence to conclude that the type of restaurant is a significant factor.

Do not reject H₀. There is sufficient evidence to conclude that the type of restaurant is a significant factor.

(f)

Predict the Consumer Reports customer satisfaction score for a seafood/steakhouse that has an average meal price of $20. (Round your answer to two decimal places.)

How much would the predicted score have changed for an Italian restaurant? (Round your answer to two decimal places.)

The predicted satisfaction score increases by points for Italian restaurants.

Answer 1

a)

Regression Statistics
Multiple R	0.5867
R Square	0.3442
Adjusted R Square	0.3097
Standard Error	3.0257
Observations	21
ANOVA
	df	SS	MS	F	Significance F
Regression	1	91.3	91.3	9.97	0.0052
Residual	19	173.9	9.1552
Total	20	265.2
	Coefficients	Standard Error	t Stat	P-value	lower 95%	upper 95%
Intercept	69.2760	3.400	20.373	0.0000	62.1588	76.39
X	0.5586	0.177	3.158	0.0052	0.1883	0.9288

so, regression line is   Ŷ =   69.28   +   0.56   *x

b)

H0: β1 = 0
Ha: β1 ≠ 0

value of F (test stat) = 9.97

p-value =    0.005

Reject H0. There is sufficient evidence to conclude that there is a significant relationship.

c)  

Let x2 = 1 if the restaurant is an Italian restaurant and x2 = 0 if the restaurant is a seafood/steakhouse.

d)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.72541938
R Square	0.526233277
Adjusted R Square	0.47359253
Standard Error	2.642189572
Observations	21
ANOVA
	df	SS	MS	F	Significance F
Regression	2	139.577112	69.78855602	9.996690909	0.001202489
Residual	18	125.6609832	6.981165733
Total	20	265.2380952
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	67.40485376	3.053452237	22.07496582	1.74112E-14	60.98978866	73.81991887	60.98978866	73.81991887
X2	3.038207841	1.155225167	2.629970268	0.016992448	0.611169827	5.465245855	0.611169827	5.465245855
X1	0.573428749	0.154574333	3.709728122	0.001603536	0.248680127	0.898177372	0.248680127	0.898177372

Y^ = 67.40 + 0.57*X1 + 3.04*X2

e)

H0: β2 = 0
Ha: β2 ≠ 0

test stat = 10.00

p value=0.001

Reject H0. There is sufficient evidence to conclude that the type of restaurant is a significant factor.

f)

Y^ = 67.40 + 0.57*20 + 3.04*0 = 78.87

The predicted satisfaction score increases by points 3.04 for Italian restaurants.