Question

A vending machine company wants to check three of its machines to determine if they are properly dispensing 12 ounces of coffee. Their data is given below α = 0.01.

Row Machine A Machine B Machine C
1 11.5 10.3 11.1
2 12.1 9.7 11.3
3 11.6 10.4 11.9
4 12.0 10.7 11.5
5 11.1 9.9 11.7
6 12.2 10.1 11.3
H0: μA = μB = μC
Ha: Not all means are equal
One-way ANOVA: Machine A, Machine B, Machine C
Source DF SS MS F P
Factor 2 8.363 4.182 31.73 0.000
Error 15 1.977 0.132
Total 17 10.340
P-value: _____
Decision: _____
Is there a significant difference between the vending machines A, B, and C? Use α=0.05.
A. No, there is no significant difference between the means.
B. Yes, there is a significant difference between the means.
C. The F-test cannot be used to answer whether or not there is a significant difference between the means.

Answer 1

The p-value of 0.000 is less than the significance level of 0.01, leading to the rejection of the null hypothesis. There is a significant difference between the vending machines' coffee dispensing volumes.

Step-by-step explanation:

You asked whether there is a significant difference between the vending machines A, B, and C in dispensing the correct amount of coffee based on the one-way ANOVA results provided. Looking at the p-value provided in the ANOVA summary table, we see the p-value is 0.000. To make a decision, we compare the p-value to the significance level α = 0.01. Since the p-value (0.000) is less than α (0.01), we reject the null hypothesis (H0: μA = μB = μC).

The decision is that there is a significant difference between at least two of the machine's mean dispensed coffee volumes. In other words, not all vending machines are dispensing the same average amount of coffee. This suggests that some machines may need calibration or maintenance to ensure they dispense the correct amount.

Answer 2

The correct option is B.

Step-by-step explanation:

The hypothesis to determine whether the vending machines are properly dispensing 12 ounces of coffee is:

H₀:
`\mu_(A)=\mu_(B)=\mu_(C)`

Hₐ: Not all means are equal.

The ANOVA output is as follows:

One-way ANOVA: Machine A, Machine B, Machine C

Source DF SS MS F P

Factor 2 8.363 4.182 31.73 0.000

Error 15 1.977 0.132

Total 17 10.340

The significance level is α = 0.05.

The p-value of the model is:

p-value = 0.000

Decision rule:

If the p-value of the test is less than the significance level then the null hypothesis will be rejected.

p-value = 0.000 < α = 0.05

The null hypothesis will be rejected.

Conclusion:

There is a significant difference between the means.

Thus, the correct option is B.