1. The goal of a nutritional study was to compare the caloric intake of adolescents living in rural areas of the United States with the caloric intake of adolescents living in urban areas of the United States. A random sample of ninth-grade students from one high school in a rural area was selected. Another random sample of ninth graders from one high school in an urban area was also selected. Each student in each sample kept records of all the food he or she consumed in one day.
The back-to-back stemplot below displays the number of calories of food consumed per kilogram of body weight for each student on that day.
\begin{tabular}{rl|l}
\multicolumn{1}{r|}{ Urban } & \multicolumn{1}{l}{ Rural } \
99998876 & 2 & \
44310 & 3 & 2334 \
97665 & 3 & 56667 \
20 & 4 & 02224 \
& 4 & 56889 \
& 5 & 1
\end{tabular}

Stem: tens
Leaf: ones
a) Write a few sentences comparing the distribution of the daily caloric intake of ninth-grade students in the rural high school with the distribution of the daily caloric intake of ninth-grade students in the urban high school.
b) Is it reasonable to generalize the findings of this study to all rural and urban ninth-grade students in the United States? Explain.
c) Researchers who want to conduct a similar study are debating which of the following two plans to use.
Plan I: Have each student in the study record all the food he or she consumed in one day. Then researchers would compute the number of calories of food consumed per kilogram of body weight for each student for that day.
Plan II: Have each student in the study record all the food he or she consumed over the same 7day period. Then researchers would compute the average daily number of calories of food consumed per kilogram of body weight for each student during that 7-day period.
Assuming that the students keep accurate records, which plan, I or II, would better meet the goal of the study? Justify your answer.

Question

1. The goal of a nutritional study was to compare the caloric intake of adolescents living in rural areas of the United States with the caloric intake of adolescents living in urban areas of the United States. A random sample of ninth-grade students from one high school in a rural area was selected. Another random sample of ninth graders from one high school in an urban area was also selected. Each student in each sample kept records of all the food he or she consumed in one day.
The back-to-back stemplot below displays the number of calories of food consumed per kilogram of body weight for each student on that day.
\begin{tabular}{rl|l}
\multicolumn{1}{r|}{ Urban } & \multicolumn{1}{l}{ Rural } \
99998876 & 2 & \
44310 & 3 & 2334 \
97665 & 3 & 56667 \
20 & 4 & 02224 \
& 4 & 56889 \
& 5 & 1
\end{tabular}

Stem: tens
Leaf: ones
a) Write a few sentences comparing the distribution of the daily caloric intake of ninth-grade students in the rural high school with the distribution of the daily caloric intake of ninth-grade students in the urban high school.
b) Is it reasonable to generalize the findings of this study to all rural and urban ninth-grade students in the United States? Explain.
c) Researchers who want to conduct a similar study are debating which of the following two plans to use.
Plan I: Have each student in the study record all the food he or she consumed in one day. Then researchers would compute the number of calories of food consumed per kilogram of body weight for each student for that day.
Plan II: Have each student in the study record all the food he or she consumed over the same 7day period. Then researchers would compute the average daily number of calories of food consumed per kilogram of body weight for each student during that 7-day period.
Assuming that the students keep accurate records, which plan, I or II, would better meet the goal of the study? Justify your answer.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To compare the distributions, we analyze the stemplot for shape, center, spread, and outliers. ‖ ‖ step 2 ⋮ The urban distribution has a higher concentration of values on the lower end, indicating a lower caloric intake per kilogram of body weight. ‖ ‖ step 3 ⋮ The rural distribution has values more spread out with a higher maximum value, suggesting a wider range and higher caloric intake per kilogram of body weight. ‖ ‖ step 4 ⋮ There are potential outliers in the urban distribution (e.g., 76) and in the rural distribution (e.g., 1). ‖ ∻Answer∻ ⚹ The urban students' caloric intake is generally lower and less varied than that of the rural students, with potential outliers present in both distributions. ⚹ ∻Key Concept∻ ⚹Comparing Distributions⚹ ∻Explanation∻ ⚹The stemplot provides a visual comparison of the distributions of caloric intake between urban and rural students, showing differences in center, spread, and potential outliers.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To determine the reasonableness of generalizing the findings, consider the sample size and selection method. ‖ ‖ step 2 ⋮ If the samples are not representative of the entire population of ninth graders in the U.S., generalization may not be reasonable. ‖ ‖ step 3 ⋮ The study uses a sample from only one high school in each area, which may not capture the diversity of the entire population. ‖ ∻Answer∻ ⚹ It is not reasonable to generalize the findings to all rural and urban ninth-grade students in the United States due to the limited and potentially unrepresentative samples. ⚹ ∻Key Concept∻ ⚹Representativeness of a Sample⚹ ∻Explanation∻ ⚹Generalizing findings to a larger population requires a sample that is representative of that population, which may not be the case in this study.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To decide which plan is better, consider the accuracy and reliability of the data collected. ‖ ‖ step 2 ⋮ Plan I involves a single day's data, which may not account for day-to-day variability in caloric intake. ‖ ‖ step 3 ⋮ Plan II averages the data over a 7-day period, which is likely to provide a more reliable estimate of typical caloric intake. ‖ ∻Answer∻ ⚹ Plan II would better meet the goal of the study as it accounts for daily variations in caloric intake and provides a more reliable estimate. ⚹ ∻Key Concept∻ ⚹Data Reliability⚹ ∻Explanation∻ ⚹Averaging data over a longer period provides a more accurate and reliable measure of typical behavior, reducing the impact of daily fluctuations.⚹◊What is the formula to compute the average daily caloric intake per kilogram of body weight for each student over a 7-day period in Plan II?⍭ Generate me a similar question◊

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