Question

Samples of six different brands of diet or imitation margarine were analyzed to determine the level of physiologically active polyunsaturated fatty acids (PAPUFA, in percent), resulting in the data shown in the accompanying table. (The data are fictitious, but the sample means agree with data reported in Consumer Reports.) $$\begin{array}{llllll} \text { Imperial } & 14.1 & 13.6 & 14.4 & 14.3 & \\ \text { Parkay } & 12.8 & 12.5 & 13.4 & 13.0 & 12.3 \\ \text { Blue Bonnet } & 13.5 & 13.4 & 14.1 & 14.3 & \\ \text { Chiffon } & 13.2 & 12.7 & 12.6 & 13.9 & \\ \text { Mazola } & 16.8 & 17.2 & 16.4 & 17.3 & 18.0 \\ \text { Fleischmann's } & 18.1 & 17.2 & 18.7 & 18.4 & \end{array}$$ a. Test for differences among the true average PAPUFA percentages for the different brands. Use \(\alpha=.05\). b. Use the T-K procedure to compute \(95 \%\) simultaneous confidence intervals for all differences between means and interpret the resulting intervals.

Short answer

Using the ANOVA test, we can understand whether there is a difference in PAPUFA percentages among the different margarine brands or not. The Tukey-Kramer procedure will then provide us with simultaneous confidence intervals to understand these differences between every pair of means. The interpretation of these intervals helps us understand where significant differences occur.

Step-by-step solution

Perform ANOVA Test

For the ANOVA test, we calculate the within group sum of squares (WSS), between group sum of squares (BSS), and the total sum of squares (TSS). Then compute the degrees of freedom and mean sum of squares (MSS). Finally, calculate the F-value and P-value. If the P-value is less than \(\alpha = 0.05\), then reject the null hypothesis.

Analyze ANOVA result

Depending on the F-value and P-value from previous step, decide whether to reject or not reject the null hypothesis 'All means are equal'. If P-value < \(\alpha\), reject the null hypothesis indicating that there are significant differences between the group means.

Calculate Tukey-Kramer Confidence Intervals

For paired comparison of means and the simultaneous confidence intervals, compute Tukey's HSD (Honest Significant Difference) value and calculate the upper and lower intervals by adding and subtracting HSD from the group mean differences to get the intervals.

Interpret Confidence Intervals

Depending on whether the interval contains 0 or not, we can interpret if there is a significant difference between the groups or not. If the interval contains 0, there is no significant difference. If not, there is a significant difference.

Key concepts

Polyunsaturated Fatty Acids Analysis

Polyunsaturated fatty acids (PUFAs) are essential fats that our bodies cannot produce and thus, must be obtained from our diet. In nutrition studies, the analysis of PUFAs within products like margarine is critical to understand their health implications.

The analysis typically involves extracting the fats from food samples and using methods like gas chromatography to quantify the levels of PUFAs. The data for different brands in our exercise give various PAPUFA percentages, which reflect the product's quality and health benefits. These variations also pave the way for statistical comparison to identify if the differences in PUFA levels are significant or merely due to chance.

Tukey-Kramer Procedure

The Tukey-Kramer (T-K) procedure is a post-hoc analysis commonly used after an ANOVA test. It’s particularly useful when you want to perform multiple pairwise comparisons between group means without increasing the Type I error rate.

The T-K method controls the familywise error rate and is robust even when the sample sizes are unequal. It calculates the Honest Significant Difference (HSD), a value that determines the range within which the true mean difference between groups lies. By comparing this range to zero, researchers can ascertain whether the differences between each pair of group means are statistically significant or not.

Simultaneous Confidence Intervals

Simultaneous confidence intervals in the context of ANOVA and the T-K procedure are intervals within which we can expect the true differences between population means to fall, for all pairwise comparisons, with a certain degree of confidence. These intervals are 'simultaneous' because they provide a confidence level for multiple comparisons at once.

For instance, a 95% simultaneous confidence interval would mean we can be 95% confident that all calculated intervals contain the true differences between the corresponding population means. Hence, these intervals are pivotal in determining not just if groups differ, but also the extent and direction of those differences.

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics that allows us to make inferences about populations based on sample data. In our exercise, the null hypothesis suggests that there is no difference among the true average PAPUFA percentages for different brands.

If the results from ANOVA show a P-value less than the chosen significance level (in this case, \( \alpha = 0.05 \)), we reject the null hypothesis, concluding that not all brand means are equal and at least one brand differs significantly from the others. Conversely, if the P-value is higher, we do not have sufficient evidence to say the brands differ regarding their PAPUFA content. This process is key to ensuring that any declared differences are not simply due to random variation.