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Chapter 15: Problem 30

The nutritional quality of shrubs commonly used for feed by rabbits was the focus of a study summarized in the article "Estimation of Browse by Size Classes for Snowshoe Hare" (Journal of Wildlife Management [1980]: 34-40). The energy content (cal/g) of three sizes (4 mm or less, \(5-7 \mathrm{~mm}\), and \(8-10 \mathrm{~mm}\) ) of serviceberries was studied. Let \(\mu_{1}, \mu_{2}\), and \(\mu_{3}\) denote the true energy content for the three size classes. Suppose that \(95 \%\) simultaneous confidence intervals for \(\mu_{1}-\mu_{2}, \mu_{1}-\mu_{3}\), and \(\mu_{2}-\mu_{3}\) are \((-10,290),(150,450)\), and \((10,310)\), respectively. How would you interpret these intervals?

Short Answer

Expert verified
The first interval indicates no significant difference in energy content between size classes 4mm or less and 5-7mm. However, the other two intervals suggest significant differences in energy content: between size class 4mm or less and size class 8-10mm, and also between size class 5-7mm and size class 8-10mm.

Step by step solution

01

Interpretation of the first interval

The first interval \((-10,290)\) is for \(\mu_{1}-\mu_{2}\). This means that we are 95% confident that the true difference in energy content between size class 4mm or less (which is represented by \(\mu_{1}\)) and size class 5-7mm (which is represented by \(\mu_{2}\)) lies within -10 and 290 cal/g. Since this interval contains zero, it implies that there is no significant difference in energy content between these two size classes.
02

Interpretation of the second interval

The second interval \((150,450)\) is for \(\mu_{1}-\mu_{3}\), comparing size class 4mm or less and size class 8-10mm. The confidence interval suggests that the true difference in energy content between these two size classes lies in the range 150 to 450 cal/g, with 95% confidence. Because this interval does not contain zero, it suggests that there is a significant difference in energy content between these two size classes.
03

Interpretation of the third interval

The third interval \((10,310)\) is for \(\mu_{2}-\mu_{3}\), comparing size class 5-7mm and size class 8-10mm. We can state with 95% confidence that the true difference in energy content between these two size classes lies within 10 and 310 cal/g. Since this interval also does not contain zero, it implies that there is a significant difference in energy content between these two size classes.

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Key Concepts

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

Statistical Inference
Statistical inference is a fundamental aspect of statistics that allows researchers to draw conclusions about populations based on sample data. It hinges on the creation and interpretation of confidence intervals and hypothesis testing.

For example, in the study of serviceberries' energy content, researchers sample different size classes and compute intervals to infer the true energy content for these populations. Constructed with a specific confidence level (in this case, 95%), these confidence intervals give us a range of values that, with a certain degree of certainty, include the true population parameter.

The essence of statistical inference lies in its ability to provide a measure of uncertainty in these estimations, helping researchers understand the likelihood that their findings reflect the true nature of the data they're examining.
Energy Content Analysis
In biological studies, energy content analysis is critical for understanding the caloric value and nutritive quality of various foods, like the serviceberries analyzed in the given study.

Assessing the energy content in calories per gram (cal/g) provides insights into the diet and feeding behavior of species, such as rabbits in this case. When evaluating energy content, it's important to consider how factors like size and age of the feed source might affect its nutritional value.

The simultaneous confidence intervals calculated in the study offer a statistical comparison among different size classes of berries, suggesting which might provide higher or equal caloric values. This analysis not only influences our understanding of hare dietary preferences but also has broader ecological implications.
Significance Testing
When intervals like those found in the serviceberry study do not include zero, significance testing is used to determine whether these differences are statistically meaningful. In this context, 'significant' doesn't refer to practical importance but indicates that it's unlikely the observed difference is due to random chance.

The confidence intervals for \(\mu_{1}-\mu_{3}\) and \(\mu_{2}-\mu_{3}\), not containing zero, suggest significant differences in energy content between these size classes with a high degree of confidence. This implies that, statistically, we can expect these differences to appear consistently across repeated samples from the population. Understanding significance testing allows scientists and statisticians to make data-driven decisions and hypotheses about their research findings.

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Most popular questions from this chapter

The article "The Soundtrack of Recklessness: Musical Preferences and Reckless Behavior Among Adolescents" (Journal of Adolescent Research [1992]: \(313-331\) ) described a study whose purpose was to determine whether adolescents who preferred certain types of music reported higher rates of reckless behaviors, such as speeding, drug use, shoplifting, and unprotected sex. Independently chosen random samples were selected from each of four groups of students with different musical preferences at a large high school: (1) acoustic/pop, (2) mainstream rock, $$\begin{array}{ccccccccc} \text { Type of Box } & & & {\text { Compression Strength (Ib) }} & & & & {\text { Sample Mean }} & {\text { Sample SD }} \\ \hline 1 & 655.5 & 788.3 & 734.3 & 721.4 & 679.1 & 699.4 & 713.00 & 46.55 \\ 2 & 789.2 & 772.5 & 786.9 & 686.1 & 732.1 & 774.8 & 756.93 & 40.34 \\ 3 & 737.1 & 639.0 & 696.3 & 671.7 & 717.2 & 727.1 & 698.07 & 37.20 \\ 4 & 535.1 & 628.7 & 542.4 & 559.0 & 586.9 & 520.0 & 562.02 & 39.87 \\ & & & & & & & \overline{\bar{x}} =682.50 & \end{array}$$ (3) hard rock, and (4) heavy metal. Each student in these samples was asked how many times he or she had engaged in various reckless activities during the last year. The following table lists data and summary quantities on driving over \(80 \mathrm{mph}\) that is consistent with summary quantities given in the article (the sample sizes in the article were much larger, but for the purposes of this exercise, we use \(\left.n_{1}=n_{2}=n_{3}=n_{4}=20\right)\) $$\begin{array}{rrrr} \text { Acoustic/Pop } & \text { Mainstream Rock } & \text { Hard Rock } & \text { Heavy Metal } \\ \hline 2 & 3 & 3 & 4 \\ 3 & 2 & 4 & 3 \\ 4 & 1 & 3 & 4 \\ 1 & 2 & 1 & 3 \\ 3 & 3 & 2 & 3 \\ 3 & 4 & 1 & 3 \\ 3 & 3 & 4 & 3 \\ 3 & 2 & 2 & 3 \\ 2 & 4 & 2 & 2 \\ 2 & 4 & 2 & 4 \\ 1 & 4 & 3 & 4 \\ 3 & 4 & 3 & 5 \\ 2 & 2 & 4 & 4 \\ 2 & 3 & 3 & 5 \\ 2 & 2 & 3 & 3 \\ 3 & 2 & 2 & 4 \\ 2 & 2 & 3 & 5 \\ 2 & 3 & 4 & 4 \\ 3 & 1 & 2 & 2 \\ 4 & 3 & 4 & 3 \\ 20 & 20 & 20 & 20 \\ 2.50 & 2.70 & 2.75 & 3.55 \\ .827 & .979 & .967 & .887 \\ 6830 & 0584 & 0351 & 7868 \end{array}$$ Also, \(N=80\), grand total \(=230.0\), and \(\overline{\bar{x}}=230.0 / 80=\) 2.875. Carry out an \(F\) test to determine if these data provide convincing evidence that the true mean number of times driving over 80 mph varies with musical preference.

The article "Utilizing Feedback and Goal Setting to Increase Performance Skills of Managers" (Academy of Management Journal \([1979]: 516-526\) ) reported the results of an experiment to compare three different interviewing techniques for employee evaluations. One method allowed the employee being evaluated to discuss previous evaluations, the second involved setting goals for the employee, and the third did not allow either feedback or goal setting. After the interviews were concluded, the evaluated employee was asked to indicate how satisfied he or she was with the interview. (A numerical scale was used to quantify level of satisfaction.) The authors used ANOVA to compare the three interview techniques. An \(F\) statistic value of \(4.12\) was reported. a. Suppose that a total of 33 subjects were used, with each technique applied to 11 of them. Use this information to conduct a level \(.05\) test of the null hypothesis of no difference in mean satisfaction level for the three interview techniques. b. The actual number of subjects on which each technique was used was \(45 .\) After studying the \(F\) table, explain why the conclusion in Part (a) still holds.

Do lizards play a role in spreading plant seeds? Some research carried out in South Africa would suggest so ("Dispersal of Namaqua Fig (Ficus cordata cordata) Seeds by the Augrabies Flat Lizard (Platysaurus broadleyi)," Journal of Herpetology [1999]: \(328-330\) ). The researchers collected 400 seeds of this particular type of fig, 100 of which were from each treatment: lizard dung, bird dung, rock hyrax dung, and uneaten figs. They planted these seeds in batches of 5 , and for each group of 5 they recorded how many of the seeds germinated. This resulted in 20 observations for each treatment. The treatment means and standard deviations are given in the accompanying table. $$\begin{array}{lccc} \text { Treatment } & \boldsymbol{n} & \overline{\boldsymbol{x}} & \boldsymbol{s} \\ \hline \text { Uneaten figs } & 20 & 2.40 & .30 \\ \text { Lizard dung } & 20 & 2.35 & .33 \\ \text { Bird dung } & 20 & 1.70 & .34 \\ \text { Hyrax dung } & 20 & 1.45 & .28 \end{array}$$ a. Construct the appropriate ANOVA table, and test the hypothesis that there is no difference between mean number of seeds germinating for the four treatments. b. Is there evidence that seeds eaten and then excreted by lizards germinate at a higher rate than those eaten and then excreted by birds? Give statistical evidence to sup- port your answer.

The article "Growth Response in Radish to Sequential and Simultaneous Exposures of \(\mathrm{NO}_{2}\) and \(\mathrm{SO}_{2} "(\) Environmental Pollution \([1984]: 303-325\) ) compared a control group (no exposure), a sequential exposure group (plants exposed to one pollutant followed by exposure to the second four weeks later), and a simultaneous-exposure group (plants exposed to both pollutants at the same time). The article states, "Sequential exposure to the two pollutants had no effect on growth compared to the control. Simultaneous exposure to the gases significantly reduced plant growth." Let \(\bar{x}_{1}, \bar{x}_{2}\), and \(\bar{x}_{3}\) represent the sample means for the control, sequential, and simultaneous groups, respectively. Suppose that \(\bar{x}_{1}>\bar{x}_{2}>\bar{x}_{3}\). Use the given information to construct a table where the sample means are listed in increasing order, with those that are judged not to be significantly different underscored. \(15.30\) The nutritional quality of shrubs commonly used for feed by rabbits was the focus of a study summarized in the article "Estimation of Browse by Size Classes for Snowshoe Hare" (Journal of Wildlife Management [1980]: \(34-40\) ). The energy content \((\mathrm{cal} / \mathrm{g})\) of three sizes \((4 \mathrm{~mm}\) or less, \(5-7 \mathrm{~mm}\), and \(8-10 \mathrm{~mm}\) ) of serviceberries was studied. Let \(\mu_{1}, \mu_{2}\), and \(\mu_{3}\) denote the true energy content for the three size classes. Suppose that \(95 \%\) simultaneous confidence intervals for \(\mu_{1}-\mu_{2}, \mu_{1}-\mu_{3}\), and \(\mu_{2}-\mu_{3}\) are \((-10,290),(150,450)\), and \((10,310)\), respectively. How would you interpret these intervals?

Leaf surface area is an important variable in plant gas-exchange rates. The article "Fluidized Bed Coating of Conifer Needles with Glass Beads for Determination of Leaf Surface Area" (Forest Science [1980]: 29-32) included an analysis of dry matter per unit surface area \(\left(\mathrm{mg} / \mathrm{cm}^{3}\right)\) for trees raised under three different growing conditions. Let \(\mu_{1}, \mu_{2}\), and \(\mu_{3}\) represent the true mean dry matter per unit surface area for the growing conditions 1 , 2 , and 3 , respectively. The given \(95 \%\) simultaneous confidence intervals are based on summary quantities that appear in the article: \(\begin{array}{llll}\text { Difference } & \mu_{1}-\mu_{2} & \mu_{1}-\mu_{3} & \mu_{2}-\mu_{3}\end{array}\) \(\begin{array}{llll}\text { Interval } & (-3.11,-1.11) & (-4.06,-2.06) & (-1.95, .05)\end{array}\) Which of the following four statements do you think describes the relationship between \(\mu_{1}, \mu_{2}\), and \(\mu_{3} ?\) Explain your choice. a. \(\mu_{1}=\mu_{2}\), and \(\mu_{3}\) differs from \(\mu_{1}\) and \(\mu_{2}\). b. \(\mu_{1}=\mu_{3}\), and \(\mu_{2}\) differs from \(\mu_{1}\) and \(\mu_{3}\). c. \(\mu_{2}=\mu_{3}\), and \(\mu_{1}\) differs from \(\mu_{2}\) and \(\mu_{3}\). d. All three \(\mu\) 's are different from one another.
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