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Chapter 27: Problem 14

To detect the presence of harmful insects in farm fields, we can put up boards covered with a sticky material and examine the insects trapped on the boards. Which colors attract insects best? Experimenters placed six boards of each of four colors at random locations in a field of oats and measured the number of cereal leaf beetles trapped. Here are the data: \(^{9}=\) III BEETLES (a) Is there evidence that the colors differ in their ability to attract beetles? Use ANOVA to answer this question and state carefully the conclusion from this ANOVA. (b) How many pairwise comparisons are there when we compare four colors? (c) Which pairs of colors are significantly different when we require significance level \(5 \%\) for all comparisons as a group? In particular, is yellow significantly better than every other color?

Short Answer

Expert verified
Yes, ANOVA shows significant differences. Six pairwise comparisons exist. Use post-hoc tests to check color differences.

Step by step solution

01

Understand the Problem

The problem involves using ANOVA to test if there are any significant differences in the means of beetle counts across boards of different colors and perform pairwise comparisons.
02

Setting up the Hypotheses for ANOVA

The null hypothesis ( H_0 ) is that all colors attract beetles equally, i.e., there is no difference in the means regardless of color. The alternative hypothesis ( H_a ) is that at least one color has a different mean from the others.
03

Perform the ANOVA Test

To conduct the ANOVA test, compute the mean square between groups and the mean square within groups. Calculate the F-statistic by dividing the mean square between by the mean square within. Use F-distribution tables to determine the critical value for the given significance level, and check if the F-statistic is greater than the critical value.
04

Analyze ANOVA Results

If the F-statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a statistically significant difference in the means of beetle counts across different colored boards.
05

Determine Number of Pairwise Comparisons

When comparing four colors, we use the formula \(\frac{k(k-1)}{2}\) where \(k\) is the number of groups. For four colors, \(\frac{4(4-1)}{2} = 6\) pairwise comparisons.
06

Conduct Pairwise Comparisons

Use post-hoc tests such as Tukey's HSD to determine which pairs of colors have significantly different means at a significance level of 5%. Check if yellow color is better than every other color specifically in these comparisons.
07

Draw Final Conclusions

Conclude based on the ANOVA results and the pairwise comparison: (a) the presence of significantly different attraction ability among the colors if applicable, (b) the significant pairs from the comparisons, and (c) whether yellow is significantly better than others.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pairwise Comparison
In statistical analysis, pairwise comparison is a method used to compare the means of two individual groups. When conducting an ANOVA test, pairwise comparisons are essential to determine which specific groups differ from each other in terms of their mean values. In this case, we are comparing different colored boards to see which attracts beetles more effectively.
  • The formula to calculate the number of pairwise comparisons among groups is \( \frac{k(k-1)}{2} \) where \( k \) is the number of groups.
  • For four colors, the number of comparisons is \( \frac{4(4-1)}{2} = 6 \).
  • Pairwise comparisons can be conducted using post-hoc tests like Tukey's Honest Significant Difference (HSD) to determine which color pairs have significantly different means.
By performing these comparisons, it is possible to identify pairs where one color stands out in successfully trapping beetles more than the other.
Statistical Hypothesis Testing
Statistical hypothesis testing is a fundamental method used to make decisions or inferences about a population based on sample data. In the context of ANOVA, hypothesis testing helps us decide if the mean number of beetles differs across different colors of boards.
  • The null hypothesis (\( H_0 \)) in this experiment asserts that all colors attract beetles equally, with no significant difference in their mean counts.
  • The alternative hypothesis (\( H_a \)) suggests that at least one color attracts a different mean number of beetles compared to others.
  • These hypotheses set the groundwork for comparisons and analysis through ANOVA tests.
Based on the test results, we can either reject the null hypothesis if a significant difference is found or fail to reject it if there is no substantial evidence against it.
F-statistic
The F-statistic is a crucial component of the ANOVA test, functioning as a means to identify differences among group means. It is calculated by taking the ratio of variance between the groups to the variance within the groups.
  • This ratio helps in determining if the variance among group means is more than what we would expect from random chance.
  • The formula for the F-statistic is \( F = \frac{\text{Mean Square Between Groups}}{\text{Mean Square Within Groups}} \).
  • By comparing the calculated F-statistic with a critical value from the F-distribution tables, we can assess the significance of our results.
If the F-statistic is greater than the critical value, the evidence suggests a significant difference in means across the groups.
Significance Level
In statistical testing, the significance level helps in determining the threshold for rejecting or accepting the null hypothesis. It reflects the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.
  • A common significance level used in testing is \( 0.05 \) or 5%, indicating a 5% risk of concluding that a difference exists when there is none.
  • In our exercise dealing with colored boards for trapping beetles, this level determines which pairs of colors are significantly different.
  • All pairwise comparisons are considered together, maintaining the overall Type I error at the set level.
Choosing the right significance level is crucial since it influences the conclusions drawn from hypothesis tests and comparisons.

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Most popular questions from this chapter

Exercise \(27.49\) describes one of the experiments done by researchers at the University of Minne-sota to demonstrate that even being reminded of money makes people more self-sufficient and less involved with other people. Here are three more of these experiments. For each experiment, which statistical test from Chapters \(21,23,25\), and 27 would you use, and why? (a) Randomly assign student subjects to money and control groups. The control group unscrambles neutral phrases, and the money group unscrambles money- oriented phrases, as described in Exercise 27.49. Then ask the subjects to volunteer to help the experimenter by coding data sheets, about five minutes per sheet. Subjects said how many sheets they would volunteer to code. Participants in the money condition volunteered to help code fewer data sheets than did participants in the control condition. (b) Randomly assign student subjects to high-money, low-money, and control groups. After playing Monopoly for a short time, the high-money group is left with \(\$ 4000\) in Monopoly money, the low-money group with \(\$ 200\), and the control group with no money. Each subject is asked to imagine a future with lots of money (high-money group), a little money (low-money group), or just their future plans (control group). Another student walks in and spills a box of 27 pencils. How many pencils does the subject pick up? "Participants in the high-money condition gathered fewer pencils" than subjects in the other two groups. (c) Randomly assign student subjects to three groups. All do paperwork while a computer on the desk shows a screensaver of currency floating underwater (Group 1), a screensaver of fish swimming underwater (Group 2), or a blank screen (Group 3). Each subject must now develop an advertisement and can choose whether to work alone or with a partner. Count how many in each group make each choice. Choosing to perform the task with a coworker was reduced among money condition participants.

As part of an ANOVA that compares three treatments, you carry out Tukey pairwise tests at the overall \(5 \%\) significance level. The Tukey tests find that \(\boldsymbol{\mu}_{1}\) is significantly different from \(\boldsymbol{\mu}_{2}\) but that the other two comparisons are not significant. You can be \(95 \%\) confident that (a) \(\boldsymbol{\mu}_{1} \neq \boldsymbol{\mu}_{2}\) and \(\boldsymbol{\mu}_{1}=\boldsymbol{\mu}_{3}\) and \(\boldsymbol{\mu}_{2}=\boldsymbol{\mu}_{3}\). (b) just \(\boldsymbol{\mu}_{1} \neq \boldsymbol{\mu}_{2}\); there is not enough evidence to draw conclusions about the other pairs of means. (c) \(\boldsymbol{\mu}_{1}=\boldsymbol{\mu}_{3}\) and \(\boldsymbol{\mu}_{2}=\boldsymbol{\mu}_{3}\), and this implies that it must also be true that \(\boldsymbol{\mu}_{1}=\boldsymbol{\mu}_{2}\).

To test whether a hearing aid is right for a patient, audiologists play a tape on which words are pronounced at low volume. The patient tries to repeat the words. There are several different lists of words that are supposed to be equally difficult. Are the lists equally difficult when there is background noise? To find out, an experimenter had subjects with normal hearing listen to four lists with a noisy background. The response variable was the percent of the 50 words in a list that the subject repeated correctly. The data set contains 96 responses. \({ }^{17}\) Here are two study designs that could produce these data: Design A. The experimenter assigns 96 subjects to four groups at random. Each group of 24 subjects listens to one of the lists. All individuals listen and respond separately. Design B. The experimenter has 24 subjects. Each subject listens to all four lists in random order. All individuals listen and respond separately. Does Design A allow use of one-way ANOVA to compare the lists? Does Design B allow use of one-way ANOVA to compare the lists? Briefly explain your answers.

Comparisons. Exercise \(27.42\) examines the relationship between previous video game experience and a surgeon's ability to acquire skills for laparoscopic surgery. Software gives these Tukey \(95 \%\) simultaneous confidence intervals: (a) How confident are you that all three of these intervals capture the true differences between pairs of population means? (b) Write a short summary of the results of the ANOVA, including the multiple comparisons.

Many states require schoolchildren to take regular statewide tests to assess their progress. Children with learning disabilities who read poorly may not do well on mathematics tests because they can't read the problems. Most states allow "accommodations" for learning-disabled children. Randomly assign 100 learning-disabled children in equal numbers to three types of accommodation and a control group: math problems are read by a teacher, math problems are read by a computer, math problems are read by a computer that also shows a video, and standard test conditions. The researcher would like to compare the mean scores on the state mathematics assessment.
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