To test for any significant difference in the number of...

To test for any significant difference in the number of hours between breakdowns for four machines, the following data were obtained. a. At the $\alpha=.05$ level of significance, what is the difference, if any, in the population mean times among the four machines? b. Use Fisher's LSD procedure to test for the equality of the means for machines 2 and 4. Use a .05 level of significance.

To test for any significant difference in the number of hours between breakdowns for four machines, the following data were obtained.
a. At the $\alpha=.05$ level of significance, what is the difference, if any, in the population mean times among the four machines?
b. Use Fisher's LSD procedure to test for the equality of the means for machines 2 and
4. Use a .05 level of significance.

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical technique used to compare the means of three or more groups to determine if at least one group mean is statistically different from the others. It does this by partitioning the total variance observed in the data into variance between groups and variance within groups, and then using the F-test to assess the significance of the differences.

F-test

The F-test is an integral part of ANOVA that compares the ratio of between-group variance to within-group variance. A large F-statistic suggests that the variability between group means is more than what would be expected by random chance, leading to the rejection of the null hypothesis.

Null Hypothesis and Significance Level

In the context of testing for differences among means, the null hypothesis typically states that there are no differences among the group means. The significance level, often set at 0.05, is the probability threshold used to decide whether an observed effect is statistically significant, balancing the risk of rejecting a true null hypothesis.

Fisher's Least Significant Difference (LSD) Procedure

Fisher's LSD is a post-hoc comparison method used after finding a significant result in an ANOVA. It allows for pairwise comparisons between group means to identify which specific groups differ from each other. While it provides a simple approach to making these comparisons, it is most reliable when the overall ANOVA indicates significant differences among groups.

Transcript

00:01 In part a, we're going to do an anova test that's going to have a test to see if our four lists have the same mean.

00:10 We'll put that null hypothesis down here in just one second.

00:13 So we're going to be assuming that the mean for machine one is equal to that of machine two is equal to that of machine three is equal to that of machine four.

00:22 And alternately, not all means are equal.

00:29 And so putting this into my ti -84, i find that my f statistic is 15, i'm sorry, 19 .99, and we're round to 5.

00:41 And this has a p value of 3 .1 times 10 to the negative 6 power, so 0 .50s and 3 .1.

00:53 And this is definitely smaller than that 0 .05 significance level.

00:58 So we have strong evidence to reject the null, so we know that we have evidence that there is a difference between at least one pair of those means.

01:07 Now on part b, we want to use the fisher's lsd test to find out, is there a difference between machine 2 and machine 4, and alternately that there is not a difference? and i found the x bar of machine 2, and that was 9 .1, and machine 4 had an x bar 11 .4.

01:36 And our mean square error, which we need for our analysis, we have 20 degrees of freedom.

01:45 And each of the sample sizes was six.

01:48 And so that degrees of freedom was that total number of numbers less than number of categories, so 20.

01:54 And our mean square air is 0 .963.

01:59 Now, our test statistic is a t value that has 20 degrees of freedom, and we take the difference between the means.

02:06 Now, if you want to take the absolute difference, you can...

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