In: Statistics and Probability
A statistical program is recommended.
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 =
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.)
Is type of restaurant a significant factor in overall customer satisfaction? (Use α = 0.05.)
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 =
(f)
Predict the Consumer Reports customer satisfaction score for a seafood/steakhouse that has an average meal price of $25. (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.
(a) the estimated regression equation is y=69.28+0.56*x1
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.586670267 | |||||
R Square | 0.344182002 | |||||
Adjusted R Square | 0.309665265 | |||||
Standard Error | 3.025748421 | |||||
Observations | 21 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 1 | 91.29017857 | 91.29018 | 9.971453 | 0.005182 | |
Residual | 19 | 173.9479167 | 9.155154 | |||
Total | 20 | 265.2380952 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 69.27604167 | 3.400463147 | 20.37253 | 2.28E-14 | 62.15879 | 76.39329 |
x1 | 0.55859375 | 0.17689553 | 3.157761 | 0.005182 | 0.188347 | 0.92884 |
(b) y=70.4431+0.5734*x1-3.0382*x2
here x2=0 for Italian and x2=1 for Seafood/Steakhouse
following regression analysis information has been generated using ms-excel and
here Italian=0 and Seafood/Steakhouse=1 as dummy variable for type(x2)
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 | 69.78856 | 9.996691 | 0.001202 | |
Residual | 18 | 125.6609832 | 6.981166 | |||
Total | 20 | 265.2380952 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 70.44306161 | 3.002376045 | 23.46244 | 6.02E-15 | 64.1353 | 76.75082 |
Price(x1) | 0.573428749 | 0.154574333 | 3.709728 | 0.001604 | 0.24868 | 0.898177 |
Type(x2) | -3.038207841 | 1.155225167 | -2.62997 | 0.016992 | -5.46525 | -0.61117 |
(third part)
Is type of restaurant a significant factor in overall customer satisfaction? (Use α = 0.05.)
answer :yes significant as the p-value=0.017 is less than alpha=0.05
Find the value of the test statistic. (Round your answer to two decimal places.) ,
answer: test statistic t=-2.63
Find the p-value. (Round your answer to three decimal places.)
p-value =0.017
(fourth part)
(a)Predict the Consumer Reports customer satisfaction score for a seafood/steakhouse that has an average meal price of $25. (Round your answer to two decimal places.)
y^=70.4431+0.5734*x1-3.0382*x2
x1=25 and for Seafood/Steakhouse x2=1 then y^=70.4431+0.5734*x1-3.0382*1=67.40
(b)How much would the predicted score have changed for an Italian restaurant? (Round your answer to two decimal places.)
x1=25 and for Italian x2=0 then y^=70.4431+0.5734*x1-3.0382*0=70.44
(c)The predicted satisfaction score increases by points for Italian restaurants.
answer is 70.44-67.40=3.04