There are 4 small franchise Quick Food Stops in the Columbus area. Numbers of burgers sold on a daily basis was recently studied. There were 6 2-month periods studied. At the 0.05 level of significance determine of there is any difference in the average number sold by region and is there any difference in the average daily sales by period
RESTAURANTS
Period Metro Interstate University River
A 124 160 320 190
B 234 220 340 230
C 430 290 290 240
D 105 245 310 170
E 240 205 280 180
F 310 260 270 205
To determine if there is any difference in the average number of burgers sold by region and by period, we can conduct a Two-Way ANOVA test. This will allow us to analyze the variation caused by the two independent variables: region and period.
Step 1: Hypotheses
We need to define our null and alternative hypotheses for both region and period.
Null Hypothesis (H0): There is no difference in the average number of burgers sold among the regions.
Alternative Hypothesis (Ha): There is a difference in the average number of burgers sold among the regions.
Null Hypothesis (H0): There is no difference in the average number of burgers sold among the periods.
Alternative Hypothesis (Ha): There is a difference in the average number of burgers sold among the periods.
Step 2: Calculate the means and variances
Next, calculate the means and variances for each region and period.
Step 3: Sum of squares
Calculate the sum of squares for both region and period using the formula:
SS_between = Σ(Σ(Xij - X..)²) / n
Where:
- Xij represents the individual observations.
- X.. represents the overall mean of all observations.
- n represents the number of observations in each group.
Step 4: Degrees of freedom
Calculate the degrees of freedom for both region and period using the formula:
df_between = number of groups - 1
df_within = total number of observations - number of groups
Step 5: Mean sum of squares
Calculate the mean sum of squares for both region and period using the formula:
MS_between = SS_between / df_between
MS_within = SS_within / df_within
Step 6: F-statistic
Calculate the F-statistic for both region and period using the formula:
F = MS_between / MS_within
Step 7: Acceptance region
Using the 0.05 level of significance, look up the critical F-value for the given degrees of freedom.
Step 8: Compare F-statistic and critical F-value
If the calculated F-statistic is greater than the critical F-value, we reject the null hypothesis and conclude that there is a significant difference in the average number of burgers sold among the regions or periods. If the calculated F-statistic is less than the critical F-value, we fail to reject the null hypothesis.
By following these steps, you can determine if there is any difference in the average number of burgers sold by region and by period in the Columbus area.