Testing Chemical Processes. To test whether the mean time...

Testing Chemical Processes. To test whether the mean time needed to mix a batch of material is the same for machines produced by three manufacturers, the Jacobs Chemical Company obtained the following data on the time (in minutes) needed to mix the material.$$ \begin{aligned} &\text { Manufacturer }\\ &\begin{array}{ccc} \mathbf{1} & \mathbf{2} & \mathbf{3} \\ 20 & 28 & 20 \\ 26 & 26 & 19 \\ 24 & 31 & 23 \\ 22 & 27 & 22 \end{array} \end{aligned} $$ a. Use these data to test whether the population mean times for mixing a batch of material differ for the three manufacturers. Use $\alpha=.05$. b. At the $\alpha=.05$ level of significance, use Fisher's LSD procedure to test for the equality of the means for manufacturers 1 and 3. What conclusion can you draw after carrying out this test?

Testing Chemical Processes. To test whether the mean time needed to mix a batch of material is the same for machines produced by three manufacturers, the Jacobs Chemical Company obtained the following data on the time (in minutes) needed to mix the material.$$
\begin{aligned}
&\text { Manufacturer }\\
&\begin{array}{ccc}
\mathbf{1} & \mathbf{2} & \mathbf{3} \\
20 & 28 & 20 \\
26 & 26 & 19 \\
24 & 31 & 23 \\
22 & 27 & 22
\end{array}
\end{aligned}
$$
a. Use these data to test whether the population mean times for mixing a batch of material differ for the three manufacturers. Use $\alpha=.05$.
b. At the $\alpha=.05$ level of significance, use Fisher's LSD procedure to test for the equality of the means for manufacturers 1 and 3. What conclusion can you draw after carrying out this test?

Key Concepts

Pairwise Comparisons

Pairwise comparisons involve individual examinations of differences between the means of each pair of groups following an overall test like ANOVA. This approach allows researchers to pinpoint exactly which pairs of groups show statistically significant differences, offering more detailed insights than the overall test that only indicates the presence of some difference among the groups.

Fisher's LSD Procedure

Fisher's Least Significant Difference (LSD) procedure is a post-hoc analysis technique used after obtaining a significant ANOVA result. It facilitates pairwise comparisons between group means by determining the smallest difference that must be exceeded for two means to be deemed significantly different. This method helps in identifying which specific groups differ from each other while keeping the error rate in check for each individual comparison.

Significance Level

The significance level, commonly denoted by ?, is the probability threshold set by the researcher for rejecting the null hypothesis. It represents the maximum acceptable probability of making a Type I error, i.e., falsely declaring a difference when none exists. Setting ? at a value like 0.05 implies that there is a 5% risk of incorrectly rejecting the null hypothesis.

One-Way ANOVA

One-way Analysis of Variance (ANOVA) is a statistical method used to compare the means of three or more independent groups to assess whether at least one group's mean differs significantly from the others. It segregates the overall variation into between-group and within-group components, using an F-test to determine if the variability among group means is greater than expected by chance under the null hypothesis that all group means are equal.

Null and Alternative Hypotheses

In the context of ANOVA, the null hypothesis asserts that there are no significant differences among the group means, meaning all populations have the same mean. The alternative hypothesis suggests that at least one group mean is different. This framework forms the basis of the statistical test to decide if observed differences can be attributed to random variation or reflect true differences in the populations.

Transcript

00:01 We'll be assuming that the mix time is the same for all three, and alternately, that they're not all equal.

00:12 And i did an anova test putting the data into the three lists of list one, list two, and list three.

00:20 And when i do that, i find my information that my f statistic is a 10 .636, an rp value associated with that.

00:30 Is very small .004 and this is definitely smaller than our 0 .05 significance level.

00:38 So we have evidence to reject the null.

00:42 So we definitely have evidence that these all the means are not equal but who is different from whom and we were asked to determine and pull out use the fisher lsd test to find out if one and three are different.

01:00 So we want to test is the mean of one and the mean of three.

01:07 Are they equal or do we have evidence that they are different? and so we need a couple pieces of information.

01:14 One is we need the mean square air.

01:17 And the mean square error from my analysis is that that is 4 .8 repeating.

01:24 And we know that our degrees of freedom, which will be the total number of numbers, less the number of categories.

01:34 So we had 12 minus 3, so we have 9.

01:37 And our test statistic t, he'll have 9 degrees of freedom.

01:41 And our denominator will be to take the mean square air and then times one over each of the sample sizes.

01:52 And my recollection is that there were four in each of these sample sizes...

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