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Nutrition Researchers Sharon Peterson and Madeleine Sigman-Grant wanted to compare the overall nutrient intake of American children (ages 2 to 19 ) who exclusively use lean meats, mixed meats, or higher-fat meats. The data given on the next page represent the daily consumption of calcium (in \(\mathrm{mg}\) ) for a random sample of eight children in each category and are based on the results presented in their article "Impact of Adopting Lower-Fat Food Choices on Nutrient Intake of American Children," Pediatrics, Vol. \(100,\) No. \(3 .\) $$ \begin{array}{ccc} \text { Lean Meats } & \text { Mixed Meats } & \text { Higher-Fat Meats } \\ \hline 844.2 & 897.7 & 843.4 \\ \hline 745.0 & 908.1 & 862.2 \\ \hline 773.1 & 948.8 & 790.5 \\ \hline 823.6 & 836.6 & 876.5 \\ \hline 812.0 & 871.6 & 790.8 \\ \hline 758.9 & 945.9 & 847.2 \\ \hline 810.7 & 859.4 & 772.0 \\ \hline 790.6 & 920.2 & 851.3 \\ \hline \end{array} $$ (a) The data was collected using a cohort observational study. Explain what this means. (b) What is the response variable in this study? (c) What are the null and alternative hypotheses in this study? (d) Test the null hypothesis that the mean calcium for each category is the same at the \(\alpha=0.05\) level of significance. Note: The requirements for a one-way ANOVA are satisfied. (e) If the null hypothesis is rejected in part (d), use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05 .\) (f) Draw boxplots of the three treatment levels to support the analytic results obtained in parts (d) and (e). (g) Can any statements of causality between meat consumption and consumption of calcium be made based on the results of this study? Explain.

Step by step solution

01

Understanding Cohort Observational Study

A cohort observational study follows a group of people who share a common characteristic over a period of time to observe outcomes. In this case, the study observes American children aged 2 to 19 who consume different types of meat and their calcium intake.

02

Identifying Response Variable

The response variable is the primary measurement of interest in a study. In this study, the response variable is the daily consumption of calcium (in mg) by American children.

03

Formulating Hypotheses

The null hypothesis (H_0) assumes that the mean calcium intake is the same across the three meat categories (lean meats, mixed meats, higher-fat meats). The alternative hypothesis (H_a) assumes that at least one of the mean calcium intakes is different. H_0: 渭_lean = 渭_mixed = 渭_high H_a: at least one 渭 is different

04

Conducting ANOVA Test

ANOVA (Analysis of Variance) is used to test if there are any statistically significant differences between the means of three or more independent groups. Since the requirements for a one-way ANOVA are satisfied, calculate the F-statistic and compare it with the critical value at 伪 = 0.05.

05

Interpreting ANOVA Results

If the p-value from the ANOVA test is less than 伪 = 0.05, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

06

Applying Tukey's Test (if necessary)

If the null hypothesis is rejected, use Tukey's HSD (Honestly Significant Difference) test to determine which specific pairs of group means are significantly different.

07

Drawing Boxplots

Create boxplots for the three categories of meat consumption (lean meats, mixed meats, higher-fat meats) to visually compare the distributions and support the analytical results from the ANOVA and Tukey's test.

08

Determining Causality

Since this study is observational and not experimental, causality cannot be established. It can only be inferred that there is an association between types of meat consumption and calcium intake.

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Key Concepts

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

nutrient intake

Nutrient intake refers to the amount of nutrients, such as vitamins, minerals, and other essential compounds, consumed through food and beverages. Nutrient intake is central to studies involving diet and health, such as the one involving American children aged 2 to 19 in this exercise. The primary focus for these researchers was to compare the calcium intake in children based on their consumption of lean meats, mixed meats, or higher-fat meats. Monitoring nutrient intake helps in understanding the impacts of dietary choices on health outcomes, including bone health, which is directly influenced by calcium intake.

Different dietary habits can significantly affect the nutrient intake. For example, children consuming lean meats might have different calcium levels compared to those consuming higher-fat meats. By calculating the daily calcium intake in milligrams (mg), we get quantitative data that can be analyzed using statistical methods like ANOVA.

cohort observational study

A cohort observational study is a type of study where a specific group of people, or cohort, is observed over a period of time to evaluate specific outcomes. In this exercise, the cohort consists of American children aged 2 to 19. These children are categorized based on their meat consumption patterns: lean meats, mixed meats, and higher-fat meats.

This study design is beneficial for understanding long-term effects and associations between dietary choices and health outcomes. However, it is important to note that while cohort studies can show associations, they do not prove causation. This means that while the study can demonstrate a relationship between meat consumption and calcium intake, it cannot definitively establish that one causes the other.

response variable

The response variable is the primary measure of interest in a study, which is affected by different conditions or groups being compared. In this observational study, the response variable is the daily consumption of calcium (measured in milligrams) among the children in the three different meat consumption categories.

Researchers focus on the response variable to understand how different diets (lean meats, mixed meats, higher-fat meats) impact the calcium intake. By analyzing the response variable, researchers can draw conclusions about dietary effects on nutrient consumption.

null hypothesis

The null hypothesis (H鈧�) is a standard assumption in statistical tests that there is no effect or no difference between groups. In this exercise, the null hypothesis is that the mean calcium intake is the same across the three categories of meat consumption: lean meats, mixed meats, and higher-fat meats.

Formally, it is expressed as:
H鈧�: \[\mu_{lean} = \mu_{mixed} = \mu_{high}\]

The alternative hypothesis (H鈧�) suggests that at least one group differs:
H鈧�: At least one \[\mu\] is different.

Testing these hypotheses using ANOVA helps determine whether the differences in calcium intake among the groups are statistically significant.

Tukey's test

Tukey's test, or Tukey's Honest Significant Difference (HSD) test, is a post-hoc analysis used when the null hypothesis in an ANOVA test is rejected. It helps in identifying which specific groups' means are significantly different from each other.

If our ANOVA test indicates significant differences in mean calcium intake among the groups, Tukey's test will provide pairwise comparisons between:

• Lean meats vs. mixed meats
• Lean meats vs. higher-fat meats
• Mixed meats vs. higher-fat meats

This helps in pinpointing where the significant differences lie, allowing for more detailed understanding of how different meat consumption affects nutrient intake across the groups.