In: Statistics and Probability
The Conch Café, located in Gulf Shores, Alabama, features casual lunches with a great view of the Gulf of Mexico. To accommodate the increase in business during the summer vacation season, Fuzzy Conch, the owner, hires a large number of servers as seasonal help. When he interviews a prospective server, he would like to provide data on the amount a server can earn in tips. He believes that the amount of the bill and the number of diners are both related to the amount of the tip. He gathered the following sample information.
Customer | Amount of Tip | Amount of Bill | Number of Diners | Customer | Amount of Tip | Amount of Bill | Number of Diners | |||||||||
1 | $ | 5.15 | $ | 74.50 | 2 | 16 | $ | 3.30 | $ | 23.59 | 2 | |||||
2 | 4.50 | 28.23 | 4 | 17 | 3.50 | 22.30 | 2 | |||||||||
3 | 1.00 | 10.65 | 1 | 18 | 3.25 | 32.00 | 2 | |||||||||
4 | 2.40 | 19.82 | 3 | 19 | 5.40 | 50.02 | 4 | |||||||||
5 | 5.00 | 28.62 | 3 | 20 | 2.25 | 17.60 | 3 | |||||||||
6 | 4.25 | 24.83 | 2 | 21 | 4.35 | 63.16 | 6 | |||||||||
7 | 0.50 | 6.25 | 1 | 22 | 3.00 | 20.27 | 2 | |||||||||
8 | 6.00 | 49.20 | 4 | 23 | 1.25 | 19.53 | 2 | |||||||||
9 | 5.00 | 43.26 | 3 | 24 | 3.25 | 27.03 | 3 | |||||||||
10 | 4.65 | 62.23 | 1 | 25 | 3.00 | 21.28 | 2 | |||||||||
11 | 5.60 | 84.81 | 1 | 26 | 6.25 | 43.38 | 4 | |||||||||
12 | 6.00 | 34.99 | 3 | 27 | 5.60 | 28.12 | 4 | |||||||||
13 | 4.00 | 33.91 | 4 | 28 | 2.50 | 26.25 | 2 | |||||||||
14 | 3.35 | 23.06 | 2 | 29 | 6.85 | 53.08 | 7 | |||||||||
15 | 0.75 | 4.65 | 1 | 30 | 8.60 | 87.85 | 8 | |||||||||
a-1. Develop a multiple regression equation with the amount of tips as the dependent variable and the amount of the bill and the number of diners as independent variables and complete the table. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)
a-2. Write out the regression equation. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)
a-3. How much does another diner add to the amount of the tips? (Round your answer to 2 decimal places.)
b-1. Complete the ANOVA table. (Leave no cells blank - be certain to enter "0" wherever required. Round "SS, MS" to 3 decimal places and "F" to 2 decimal places.)
b-2. What is your decision regarding the null-hypothesis?
c-1. Conduct an individual test on each of the variables. What is the decision rule at the 0.05 level of significance? (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)
c-2. Which variable should be deleted?
a-1)
The regression equation is defined as,
Now, the regression analysis is done in excel by following these steps
Step 1: Write the data values in excel. The screenshot is shown below,
Step 2: DATA > Data Analysis > Regression > OK. The screenshot is shown below,
Step 3: Select Input Y Range: 'y' column, Input X Range: 'x1 and x2' column then OK. The screenshot is shown below,
The result is obtained. The screenshot is shown below,
a-2)
The regression equation is,
a-3)
The estimate of the variable number of dinner = 0.4495
For one dinner increase in the number of dinners, the tip amount increases by $0.4495.
b-1)
From the regression analysis summary,
ANOVA | |||||
df | SS | MS | F | Significance F | |
Regression | 2 | 78.572 | 39.286 | 39.94 | 8.58E-09 |
Residual | 27 | 26.559 | 0.984 | ||
Total | 29 | 105.132 |
b-2)
Overall Significance
From the ANOVA table,
F | Significance F | |
Regression | 39.9380 | 8.58E-09 |
Since the significance F value is 8.58E-09 which is less than 0.05 at a 5% significance level, the null hypothesis is rejected which means the model fits the data value at the 5% significance level compared to the intercept only model.
c-1)
From, the result summary,
P-value | Significance level | |||
x1 | 1.14E-05 | < | 0.05 | Significant |
x2 | 0.001022 | < | 0.05 | Significant |
The P-value for both independent variables is less than 0.05 at a 5% significance level hence we can conclude that independent variables are statistically significant in the model.
c-2)
No variable should be deleted. (Both variables are statistically significant)
From the regression output summary,
R Square | 0.75 |