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Chapter 15: Problem 23

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.

Short Answer

Expert verified
The correct statement is 'c. \(\mu_{2} = \mu_{3}\) and \(\mu_{1}\) differs from \(\mu_{2}\) and \(\mu_{3}\)'.

Step by step solution

01

Interpret Confidence Intervals

In statistics, a confidence interval (CI) is a type of estimate, derived from the statistics of the observed data, that might contain the true value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval. More simply, the CI quantifies the level of uncertainty around a sample estimate of a population parameter. Here, the confidence level is 95%, indicating we can be 95% confident that the true population parameter lies within the interval.
02

Analyze Each Interval

In the intervals presented, \((\mu_{1} - \mu_{2})\) lies between -3.11 and -1.11, \((\mu_{1} - \mu_{3})\) lies between -4.06 and-2.06, and \((\mu_{2} - \mu_{3})\) lies between -.195 and 0.05. This tells us that \(\mu_{1}\) is smaller than both \(\mu_{2}\) and \(\mu_{3}\), because the differences with them are negative. On the other hand, the difference between \(\mu_{2}\) and \(\mu_{3}\) includes 0, meaning there is a chance that they could be the same. The confidence interval \(-1.95\) to \(0.05\) includes \(0\) in its range, meaning there is not a significant difference between the two.
03

Choose the correct statement

From the above analysis, it is clear that \(\mu_{2}\) and \(\mu_{3}\) are likely to be equal (or at least not significantly different), and \(\mu_{1}\) is different from both \(\mu_{2}\) and \(\mu_{3}\). Among the provided options, option c. \(\mu_{2} = \mu_{3}\) and \(\mu_{1}\) differs from \(\mu_{2}\) and \(\mu_{3}\) matches this finding.

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Most popular questions from this chapter

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