Math Problem Statement
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How long it takes paint to dry can have an impact on the production capacity of a business. An auto body & paint business invested in a paint-drying robot to speed up its process. An interesting question is, "Do all paint-drying robots have the same drying time?" To test this, suppose we sample five drying times for each of different brands of paint-drying robots. The time in minutes until the paint was dry enough for a second coat to be applied was recorded. Suppose the following data were obtained.
Robot 1 Robot 2 Robot 3 Robot 4
129 143 134 149
137 133 142 143
134 141 138 136
123 146 136 139
142 132 130 153
At the πΌ = 0.05 level of significance, test to see whether the mean drying time is the same for each type of robot.
State the null and alternative hypotheses.
H0: At least two of the population means are equal.
Ha: At least two of the population means are different.
H0: π1 β π2 β π3 β π4
Ha: π1 = π2 = π3 = π4
H0: π1 = π2 = π3 = π4
Ha: Not all the population means are equal.
H0: Not all the population means are equal.
Ha: π1 = π2 = π3 = π4
H0: π1 = π2 = π3 = π4
Ha: π1 β π2 β π3 β π4
Correct: Your answer is correct.
Find the value of the test statistic. (Round your answer to two decimal places.)
2.53
Incorrect: Your answer is incorrect. Find the p-value. (Round your answer to four decimal places.) p-value = .08
Solution
The value of the test statistic is approximately 2.73 (rounded to two decimal places). The p-value is approximately 0.0781 (rounded to four decimal places).
Since the p-value is greater than the significance level , we fail to reject the null hypothesis. This means there isn't sufficient evidence to suggest that the mean drying times for the different robots are significantly different.
Would you like further details or have any questions?
Here are 5 related questions to expand your understanding:
- What are the assumptions of ANOVA, and how are they tested?
- How does the significance level affect the decision in hypothesis testing?
- Can we use a post hoc test after failing to reject the null hypothesis?
- What are other ways to increase the power of an ANOVA test?
- How does sample size influence the result of the ANOVA test?
Tip: In ANOVA, it's important to ensure that variances across groups are approximately equal. This is called the homogeneity of variance assumption.
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Math Problem Analysis
Mathematical Concepts
ANOVA (Analysis of Variance)
Hypothesis Testing
Significance Level
Formulas
F-statistic = (Between-group variability) / (Within-group variability)
p-value calculation based on F-distribution
Theorems
Central Limit Theorem
Law of Large Numbers
Suitable Grade Level
Undergraduate Statistics or Business Analytics
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