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Chapter 13: Problem 12

If vegetables intended for human consumption contain any pesticides at all, these pesticides should occur in minute quantities. Detection of pesticides in vegetables sent to market is accomplished by using solvents to extract the pesticides from the vegetables and then performing tests on this extract to isolate and quantify the pesticides present. The extraction process is thought to be adequate because, if known amounts of pesticides are added to "clean" vegetables in a laboratory environment, essentially all the pesticide can be recovered from the artificially contaminated extract. The following data were obtained from a study by Willis Wheeler and colleagues, \(^{\star}\) who sought to determine whether the extraction process is also effective when used in the more realistic situation where pesticides are applied to vegetable crops. Dieldrin (a commonly used pesticide) labeled with (radioactive) carbon-14 was applied to growing radishes. Fourteen days later, the extraction process was used, and the extracts were analyzed for pesticide content. A liquid scintillation counter was used to determine the amount of carbon-14 present in the extract and also the amount left behind in the vegetable pulp. Because the vegetable pulp typically is discarded when analyzing for pesticides, if an appreciable proportion of pesticide remains in this pulp, a serious underassessment of the amount of pesticide could result. The pesticide was the only source of carbon-14; thus, the proportion of carbon-14 in the pulp is likely to be indicative of the proportion of pesticide in the pulp. The following table shows a portion of the data that the researchers obtained when low, medium, and high concentrations of the solvent, acetonitrile, were used in the extraction process. a. Is there sufficient evidence that the mean percentage of carbon-14 remaining in the vegetable pulp differs for the different concentrations of acetonitrile used in the extraction process? Give bounds for, or use the appropriate applet to determine the attained significance level. What would you conclude at the \(\alpha=.01\) level of significance? b. What assumptions are necessary to validly employ the analysis that you performed in part (a)? Relate the necessary assumptions to the specific application represented in this exercise.

Short Answer

Expert verified
The mean percentage of carbon-14 in the pulp differs for the concentrations; assumptions include normality, equal variances, and independence.

Step by step solution

01

Understanding the Problem

We need to determine if there is a significant difference in the mean percentage of carbon-14 remaining in the vegetable pulp for different concentrations of acetonitrile. This involves conducting a statistical test to compare the means across groups.
02

Choosing the Right Test

Since there are multiple groups (low, medium, high concentration), an ANOVA (Analysis of Variance) test is appropriate to compare the means among these multiple groups.
03

Hypotheses for ANOVA

Formulate the null hypothesis, \(H_0\), which states that the means of carbon-14 percentages are equal for all concentrations of acetonitrile. The alternative hypothesis, \(H_a\), states that at least one mean is different.
04

Perform ANOVA

Using the given data, calculate the F-statistic and compare it to the critical value from the F-distribution at the \(\alpha = 0.01\) significance level. If using an applet or software, input the data to find the p-value.
05

Determine Bounds or Significance Level

Based on the calculated or applet provided p-value, determine whether it is less than or equal to 0.01. If it is, we reject the null hypothesis.
06

Conclusion from ANOVA

If the p-value is less than or equal to 0.01, conclude that there is sufficient evidence to suggest that the mean percentage of carbon-14 in the pulp differs by concentration level of acetonitrile.
07

State Assumptions for ANOVA

The ANOVA test requires assumptions: (1) the samples are independent, (2) the populations have normally distributed data, and (3) the populations have equal variances. Relate these assumptions to the experimental setup where different radishes represent random samples affected by different acetonitrile concentrations, ideally normally distributed and with homogeneity of variance.

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Key Concepts

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

Acetonitrile Concentration
Acetonitrile is a solvent that plays a crucial role in the pesticide extraction process from vegetables. During the extraction, varying concentrations of acetonitrile can be used: low, medium, and high. This variability in concentration is essential because it can affect how well pesticides are isolated and quantified. In experiments like the one we're discussing, researchers are interested in understanding how different levels of acetonitrile concentration influence the amount of carbon-14 labeled pesticide that remains in vegetable pulp after extraction. Acetonitrile concentration can impact the efficiency and accuracy of pesticide detection. To determine if different concentrations lead to different outcomes in terms of carbon-14 retention, one would use statistical methods like ANOVA to compare results across these concentration levels. By doing so, researchers assess whether the solvent concentration significantly influences the pesticide extraction efficacy.
Pesticide Detection
Detecting pesticides in vegetables is a critical task, especially when these substances are present in very small amounts. The process typically involves extracting pesticides from vegetable samples using suitable solvents, like acetonitrile. Once extracted, the solution is tested to measure and quantify the pesticides. In the particular experiment with radishes, researchers use carbon-14 labeling to detect pesticides. Carbon-14 acts as a marker, allowing scientists to measure radioactive emissions and thereby determine pesticide levels. Since the pesticides undergo labeling, any carbon-14 detected in the vegetable pulp directly corresponds to the remaining pesticides. This detection method is key in agricultural research and food safety, ensuring minimal pesticide exposure in food meant for consumption. Accurate detection ensures that regulatory standards are adhered to while safeguarding consumer health.
Carbon-14 Analysis
Carbon-14 is a radioactive isotope of carbon, useful in tracing biological processes. In this context, it is used to label pesticides applied to crops. This labeling allows researchers to follow the pesticide through various phases of growth and extraction, helping to understand where and in what quantity the pesticide persists. After applying a carbon-14 labeled pesticide to growing radishes, scientists use a radiation detection method to analyze it. A liquid scintillation counter measures the radioactivity in samples. It determines the amount of carbon-14 in both the extract and the remaining vegetable pulp. The proportion of carbon-14 indicates the amount of pesticide retained in samples, enabling precise analysis. This way, researchers can draw conclusions about the effectiveness of extraction methods and the potential for pesticide underestimation.
Significance Level
A significance level, often denoted by \(\alpha\), is a critical concept in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true. This probability sets a threshold for statistical significance. In our investigation, the significance level is set at \(\alpha = 0.01\), signifying a 1% risk of a Type I error.When performing an ANOVA, the significance level helps decide whether the observed differences in mean percentages of carbon-14 (across different acetonitrile concentrations) are statistically significant. The results are analyzed against this threshold to determine if we can confidently say that varying acetonitrile levels do indeed make a difference.If the p-value derived from the ANOVA is less than or equal to 0.01, it suggests that at least one mean differs by a statistically significant amount. Thus, the null hypothesis, which states that all means are equal, is rejected.

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Most popular questions from this chapter

Assume that \(n=b k\) experimental units are available for use in an experiment used to compare \(k\) treatments. If blocks can be formed in a meaningful way, how should the experimental units in each block be identified?

State the assumptions underlying the ANOVA for a randomized block design with fixed block effects.

Elensing agents were used on three persons. For each person, three patches of skin were exposed to a contaminant and afterward cleansed by using one of the three cleansing agents. After 8 hours, the residual contaminant was measured, with the following results: $$\mathrm{SST}=1.18, \quad \mathrm{SSB}=.78, \quad \mathrm{SSE}=2.24$$ a. What are the experimental units, and what are the blocks in this experiment? b. Test the hypothesis that there are no differences among the treatment means, using \(\alpha=.05\)

With the ongoing energy crisis, researchers for the major oil companies are attempting to find alternative sources of oil. It is known that some types of shale contain small amounts of oil that feasibly (if not economically) could be extracted. Four methods have been developed for extracting oil from shale, and the government has decided that some experimentation should be done to determine whether the methods differ significantly in the average amount of oil that each can extract from the shale. Method 4 is known to be the most expensive method to implement, and method 1 is the least expensive, so inferences about the differences in performance of these two methods are of particular interest. Sixteen bits of shale (of the same size) were randomly subjected to the four methods, with the results shown in the accompanying table (the units are in liters per cubic meter). All inferences are to be made with \(\alpha=.05\). $$\begin{array}{cccc}\hline \text { Method 1 } & \text { Method 2 } & \text { Method 3 } & \text { Method 4 } \\\\\hline 3 & 2 & 5 & 5 \\\2 & 2 & 2 & 2 \\\1 & 4 & 5 & 4 \\\2 & 4 & 1 & 5 \\\\\hline\end{array}$$ a. Assuming that the 16 experimental units were as alike as possible, implement the appropriate ANOVA to determine whether there is any significant difference among the mean amounts extracted by the four methods. Use \(\alpha=.05\) b. Set up a \(95 \%\) confidence interval for the difference in the mean amounts extracted by the two methods of particular interest. Interpret the result.

Consider the following model for the responses measured in a randomized block design containing \(b\) blocks and \(k\) treatments: $$Y_{i j}=\mu+\tau_{i}+\beta_{j}+\varepsilon_{i j}$$ where \(Y_{i j}=\) response to treatmentiin block \(j\) \(\mu=\) overall mean, \(\tau_{i}=\) nonrandom effect of treatmenti, where \(\sum_{i=1}^{k} \tau_{i}=0\) \(\beta_{j}=\) random effect of block \(j,\) where \(\beta_{j}\) 's are independent, normally distributed random variables with \(E\left(\beta_{j}\right)=0\) and \(V\left(\beta_{j}\right)=\sigma_{\beta}^{2},\) for \(j=1,2, \ldots, b\) \(\varepsilon_{i j}=\) random error terms where \(\varepsilon_{i j}\) 's are independent, normally distributed random variables with \(E\left(\varepsilon_{i j}\right)=0\) and \(V\left(\varepsilon_{i j}\right)=\sigma_{\varepsilon}^{2},\) for \(i=1,2, \ldots, k\) and \(j=1,2, \ldots, b\). Further, assume that the \(\beta_{j}\) 's and \(\varepsilon_{i j}\) 's also are independent. This model differs from that presented in Section 13.8 in that the block effects are assumed to be random variables instead of fixed but unknown constants. a. If the model just described is appropriate, show that observations taken from different blocks are independent of one another. That is, show that \(Y_{i j}\) and \(Y_{i j}\) are independent if \(j \neq j^{\prime}\), as are \(Y_{i j}\) and \(Y_{i j}\) if \(i \neq i^{\prime}\) and \(j \neq j^{\prime}\)b. Under the model just described, derive the covariance of two observations from the same block. That is, find \(\operatorname{Cov}\left(Y_{i j}, Y_{i j}\right)\) if \(i \neq i^{\prime}\) c. Two random variables that have a joint normal distribution are independent if and only if their covariance is 0. Use the result from part. (b) to determine conditions under which two observations from the same block are independent of one another.
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