Question

The driver of a diesel-powered automobile decided to test the quality of three types of diesel fuel sold in the area based on mpg. Test the null hypothesis that the three means are equal using the following data. Make the usual assumptions and take \(\alpha=0.05\). $$ \begin{array}{llllll} \text { Brand A: } & 38.7 & 39.2 & 40.1 & 38.9 & \\ \text { Brand B: } & 41.9 & 42.3 & 41.3 & & \\ \text { Brand C: } & 40.8 & 41.2 & 39.5 & 38.9 & 40.3 \end{array} $$

Short answer

To adequately answer this exercise using a one-way ANOVA, several steps must be executed. These include the calculation of group means, total and group sum of squares, mean sum of squares, the F-statistic, and finally a comparison with the F-distribution. From this analysis, a conclusion is made determining whether there is a statistically significant difference in MPG among the three brands of diesel.

Step-by-step solution

Calculating Group Means

First, find the mean MPG for each brand. This can be done by summing the MPG values for each brand and dividing by the number of observations. For Brand A, the mean is \((38.7 + 39.2 + 40.1 + 38.9) / 4 = 39.225\). For Brand B, the mean is \((41.9 + 42.3 + 41.3) / 3 = 41.833\), and for Brand C, the mean is \((40.8 + 41.2 + 39.5 + 38.9 + 40.3) / 5 = 40.14\).

Calculating Total and Group Sum of Squares

Compute the sum of squares within groups (SSW) and the sum of squares between groups (SSB). SSW is calculated by summing the squares of the differences between each observation and its group mean. SSB is calculated by summing the product of the group size and the squared difference between group mean and grand mean.

Calculating Mean Sum of Squares and F-Statistic

Next, calculate the mean sum of squares within groups (MSW) and between groups (MSB). MSW is equal to SSW divided by the total degrees of freedom (dfW), and MSB is SSB divided by the degrees of freedom for groups (dfB). The F-statistic is the ratio MSB/MSW.

Comparison with the F-Distribution

Refer to the F-distribution table with dfB and dfW to determine the critical value of F at the 0.05 significance level. If the computed F-statistic is greater than the critical value, reject the null hypothesis.

Conclusion

Based on whether the null hypothesis was rejected or retained, conclude if there is a significant difference in MPG among the three diesel brands.

Key concepts

Null Hypothesis Testing

Null Hypothesis Testing is a fundamental concept in statistics used to determine whether there is sufficient evidence to reject a stated null hypothesis. In the context of the diesel fuel efficiency problem, the null hypothesis posits that there is no significant difference in miles per gallon (MPG) performance among the three brands of diesel fuel tested. To test this, an Analysis of Variance (ANOVA) is employed, which compares the variance within each group to the variance between the groups. If the ANOVA results in an F-statistic value larger than the critical value from an F-distribution (given a significance level of \( \alpha = 0.05 \)), the null hypothesis is rejected, suggesting that at least one brand's MPG performance is significantly different from the others.

In practice, this test helps discern whether observed differences in fuel efficiency are likely due to genuine brand discrepancies or random chance. In educational settings, understanding this process reinforces the importance of statistical tests in research and data-driven decision-making.

Group Means Calculation

Calculating group means is an essential step in ANOVA and involves averaging the data for each separate group. In the diesel fuel efficiency study, the aim is to measure and compare the average MPG for each brand of diesel fuel. This is accomplished by adding the MPG values for all vehicles in a group (here, a specific brand) and dividing by the number of vehicles tested. The formula for Brand A’s mean, for instance, is \( (38.7 + 39.2 + 40.1 + 38.9) / 4 = 39.225 \). These calculations serve as the basis for comparing group performance against each other and against the overall average or grand mean.

Grasping the approach to calculate group means aids students in understanding how individual data points contribute to broader patterns and inferences in statistical analysis and research.

Sum of Squares Computation

The Sum of Squares computation is a critical part of ANOVA which helps in quantifying the variability within and between groups. Specifically, it involves two components: the sum of squares within groups (SSW) and the sum of squares between groups (SSB).

Sum of Squares Within Groups (SSW)

SSW is the sum of the squared deviations of each individual observation from their group mean. It measures the variability within each group of data.

Sum of Squares Between Groups (SSB)

SSB measures the variability between the group means and the overall mean (grand mean). It is computed by multiplying the number of observations in each group by the square of the difference between that group's mean and the grand mean. These computations are foundational in determining how much of the overall variation is due to differences within groups versus differences between the groups.

F-Statistic

The F-statistic is a ratio that emerges from the ANOVA and is key in determining whether observed group differences are statistically significant. It is calculated by dividing the mean sum of squares between groups (MSB) by the mean sum of squares within groups (MSW). The MSB is the sum of squares between groups (SSB) divided by the degrees of freedom for groups (dfB), and MSW is the sum of squares within groups (SSW) divided by the total degrees of freedom within groups (dfW).

The F-statistic is guided by the principle that if group means are not dramatically different, MSB and MSW should be relatively similar, thus, the F-statistic would approach 1. A large F-statistic, suggesting a larger MSB relative to MSW, increases the evidence against the null hypothesis. Understanding how to compute and interpret the F-statistic is crucial as it is a gateway to making data-driven conclusions in many scientific fields.

F-Distribution

The F-distribution is a probability distribution that outlines how statistically significant the F-statistic calculated from an ANOVA test is. It is an asymmetric distribution that is used only for positive values, and its shape varies depending on two types of degrees of freedom—those associated with the between-group variance and those associated with the within-group variance.

After calculating the F-statistic during an ANOVA test, researchers consult F-distribution tables—at a specific significance level (e.g., \( \alpha = 0.05 \) for a 95% confidence level)—to find the critical value. If the calculated F-statistic surpasses this critical value, it indicates that the experimentally observed variance between groups is unlikely to have occurred due to chance alone, thereby warranting rejection of the null hypothesis.

Understanding the F-distribution is paramount for students as it is instrumental in interpreting the results of various statistical tests, including ANOVA, and in discerning the reliability of these results in the context of empirical research.