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There are four auto body shops in a community and all claim to promptly serve customers. To check if there is any difference in service, customers are randomly selected from each repair shop and their waiting times in days are recorded. The output from a statistical software package is: $$\begin{array}{|lclll|}\hline {\text { Summary }} \\ \text { Groups } & \text { Count } & \text { Sum } & \text { Average } & \text { Variance } \\\\\hline \text { Body Shop A } & 3 & 15.4 & 5.133333 & 0.323333 \\\\\text { Body Shop B } & 4 & 32 & 8 & 1.433333 \\\\\text { Body Shop C } & 5 & 25.2 & 5.04 & 0.748 \\\\\text { Body Shop D } & 4 & 25.9 & 6.475 & 0.595833 \\\\\hline\end{array}$$ $$\begin{array}{|lcrccc|}\hline {\text { ANOVA }} \\\\\hline \text { Source of Variation } & \text { SS } &\text { df } & \text { MS } & \text { F } & \text { p-value } \\\\\hline \text { Between Groups } & 23.37321 & 3 & 7.791069 & 9.612506 & 0.001632 \\\\\text { Within Groups } & 9.726167 & 12 & 0.810514 & & \\\\\text { Total } & 33.09938 & 15 & & & \\\\\hline\end{array}$$ Is there evidence to suggest a difference in the mean waiting times at the four body shops? Use the .05 significance level.

Step by step solution

01

Understanding the Problem

The problem involves determining if there is a significant difference in the mean waiting times at four different auto body shops based on the provided ANOVA table with a significance level of \( \alpha = 0.05 \).

02

Identify the Hypotheses

The null hypothesis \( H_0 \) states that there is no difference in the mean waiting times among the four body shops. The alternative hypothesis \( H_a \) states that at least one shop has a different mean waiting time.

03

Examine the ANOVA Table

The ANOVA table provides the calculated F-value as 9.612506 and the p-value as 0.001632 for the test comparing the means of the four groups.

04

Decision Rule

If the p-value is less than the significance level (\( \alpha = 0.05 \)), we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.

05

Conclusion

The p-value (0.001632) is less than the significance level (0.05), therefore, we reject the null hypothesis. There is enough statistical evidence to suggest a difference in the mean waiting times at the four body shops.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a fundamental part of statistics and is used to determine whether there is enough evidence in data to infer that a condition is true for a whole population. Imagine you want to test if all auto body shops deliver similar waiting times. This can be formulated into a hypothesis test.

We start by defining two hypotheses based on the research question. The null hypothesis, denoted as \( H_0 \), typically states that there is no effect or difference. In this scenario, \( H_0 \) is that the waiting times at all body shops are the same. The alternative hypothesis, \( H_a \), states the opposite, that at least one body shop's waiting time is different.

To test these hypotheses, we collect data, analyze it and assess whether the findings are consistent with the null hypothesis using statistical algorithms like ANOVA (Analysis of Variance). This serves as a pivotal part of making an informed decision. If the evidence suggests rejection of \( H_0 \), we conclude there are differences in waiting times across the shops.

P-Value

The p-value is a crucial metric in hypothesis testing, giving us a measure of the strength of evidence against the null hypothesis. Think of it as indicating the probability of observing the results we obtained, or more extreme, assuming that the null hypothesis is true.

In the example with the body shops, the p-value was calculated to be 0.001632. A very small p-value like this suggests that if \( H_0 \) were true (i.e., there is no difference in waiting times), obtaining such an extreme result would be highly unlikely.

Since the p-value is significantly low, it indicates a strong evidence against the null hypothesis. In practical scenarios, a low p-value implies that the observed data could not easily occur by chance, advocating for the acceptance of the alternative hypothesis.

Significance Level

The significance level, denoted by \( \alpha \), is a threshold set by the researcher to decide when to reject the null hypothesis. It represents the probability of making a Type I error, which is when the null hypothesis is wrongly rejected.

In our body shop example, we use a significance level of 0.05. This means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis.

• If our p-value is less than 0.05, we reject the null hypothesis, suggesting a significant difference in waiting times.
• If the p-value is greater than or equal to 0.05, we do not reject the null hypothesis.

Significance levels must be chosen before conducting the test to avoid biases. They help in maintaining objectivity and reliability in statistical analysis.

F-Value

The F-value is a statistic used in ANOVA tests to determine if the means between multiple groups are significantly different. Essentially, it compares the variance between the groups to the variance within the groups.

In the example, the F-value is calculated to be 9.612506. A high F-value indicates that there is more variability between the group means than within the groups, suggesting a potential difference among them.

• F-value is computed from the data and compared against a critical value from F-distribution tables, which depend on the chosen significance level \( \alpha \) and degrees of freedom.
• If the calculated F-value is larger than the critical value, it implies a significant difference between group means, leading to the rejection of the null hypothesis.

The F-value provides a formal assessment in ANOVA tests, thereby confirming whether the variations in group means are statistically significant.