Suppose that samples of six different brands of diet or...

Suppose that 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 accompanying data. $$ \begin{array}{llllll} \text { Imperial } & 14.1 & 13.6 & 14.4 & 14.3 & \\ \text { Parkay } & 12.8 & 12.5 & 13.4 & 13.0 & 12.3 \\ \text { Blue Bonnet } & 13.5 & 13.4 & 14.1 & 14.3 & \\ \text { Chiffon } & 13.2 & 12.7 & 12.6 & 13.9 & \\ \text { Mazola } & 16.8 & 17.2 & 16.4 & 17.3 & 18.0 \\ \text { Fleischmann's } & 18.1 & 17.2 & 18.7 & 18.4 & \end{array} $$ a. Carry out a test to determine if there is evidence of differences among the true mean PAPUFA percentages for the different brands. Use $\alpha=0.05$. b. Use the T-K procedure to calculate $95 \%$ simultaneous confidence intervals for all differences between pairs of means and give the corresponding underscoring pattern.

Suppose that 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 accompanying data.
$$
\begin{array}{llllll}
\text { Imperial } & 14.1 & 13.6 & 14.4 & 14.3 & \\
\text { Parkay } & 12.8 & 12.5 & 13.4 & 13.0 & 12.3 \\
\text { Blue Bonnet } & 13.5 & 13.4 & 14.1 & 14.3 & \\
\text { Chiffon } & 13.2 & 12.7 & 12.6 & 13.9 & \\
\text { Mazola } & 16.8 & 17.2 & 16.4 & 17.3 & 18.0 \\
\text { Fleischmann's } & 18.1 & 17.2 & 18.7 & 18.4 &
\end{array}
$$
a. Carry out a test to determine if there is evidence of differences among the true mean PAPUFA percentages for the different brands. Use $\alpha=0.05$.
b. Use the T-K procedure to calculate $95 \%$ simultaneous confidence intervals for all differences between pairs of means and give the corresponding underscoring pattern.

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare means across multiple groups to determine if at least one group mean is significantly different from the others. It decomposes total variance into between-group and within-group variability and uses an F-test to assess the evidence against the null hypothesis that all group means are equal.

Multiple Comparison Procedures

Multiple comparison procedures are techniques used after an ANOVA has indicated significant differences among group means. They adjust for the increased risk of Type I error that occurs when making several pairwise comparisons simultaneously, ensuring that the overall confidence level is maintained.

Tukey-Kramer Procedure

The Tukey-Kramer procedure is a specific multiple comparison method that provides simultaneous confidence intervals for all pairwise differences between group means. It is especially useful when sample sizes may be unequal and helps identify which specific groups differ from each other while controlling the familywise error rate.

Simultaneous Confidence Intervals

Simultaneous confidence intervals are intervals constructed for multiple parameters (such as pairwise differences between means) in such a way that the prescribed confidence level applies to the entire set of intervals. This approach accounts for the multiple comparisons being made, reducing the likelihood of Type I errors across all intervals.

Transcript

00:01 In this exercise, we are provided with the data for a sample of size 6.

00:07 And for part a, we were asked to first check the assumption that the population is normally distributed.

00:15 So we can run a check based on the sample data.

00:19 And we can do this by constructing a normal probability plot.

00:24 I've done this in r, so i will show you the steps that i used in r to do this.

00:29 So the first step is to enter the sample data.

00:31 So i put it in a variable called x.

00:34 So all of this data here is the data provided in the question.

00:39 And then i use the function qq norm and the argument x.

00:45 And that makes a normal probability plot of the sample data.

00:51 If the sample data is normally distributed, then you see a fairly straight line from the bottom left to the top right, which is the case here.

00:59 So the assumption of normality of the underlying population is reasonable.

01:28 Now, here when i talk about the underlying population, for the specific question, it is the polyunsaturated fatty acid levels in the margin of a particular brand, the diet margin of a particular brand.

01:44 So the underlying population is all the margin that that brand makes.

01:51 For b, we are asked to find a 90 % confidence interval for the population mean and provide an interpretation of this interval.

02:02 Now, if we don't know the standard deviation of the population, which is the case here, a confidence interval for the population mean can be given by this formula, and we can calculate most of these parameters from the sample data.

02:21 So first, the sample average.

02:24 This can be found simply by using the mean function and using x as the argument.

02:30 It's 16 .983.

02:40 The sample size is 6.

02:44 And the sample of standard deviation, it can be found by using the function sd and the argument x.

02:54 We get 0 .3189.

03:03 The only thing left is the relevant percentile...

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