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Chapter 10: Problem 97

It is fairly well known that perception of weight by adolescents does not always agree with actual weight. What is less clear is whether perception of weight differs by gender. For this purpose, a study was performed among students in a local high school, where students provided their actual height and weight by self-report. The following data were obtained from 286 students \((143 \text { boys and } 143\) girls). (The data for this problem were provided by Laura Rosner.) The students were classified as underweight if their bodymass index (BMI) \(\left(\mathrm{kg} / \mathrm{m}^{2}\right)\) was less than \(18.0 \mathrm{kg} / \mathrm{m}^{2}\), as normal if their BMI was \(\geq 18.0\) and \(<25.0,\) and overweight if their BMI was \(\geq 25.0\). Based on these criteria, 17 of the girls were underweight, 113 were of normal weight, and 13 were overweight. For the boys, 7 were underweight, 115 were of normal weight, and 21 were overweight. What test procedure can be used to assess whether the weight status of boys significantly differs from girls?

Short Answer

Expert verified
Use a Chi-square test for independence to assess weight status differences between boys and girls.

Step by step solution

01

Understanding the Problem

The exercise involves comparing weight perception and actual weight by gender, specifically using Body Mass Index (BMI) data for 143 boys and 143 girls. We need to determine if there is a significant difference in weight status between the genders, classified as underweight, normal, or overweight.
02

Data Summarization

The data is categorized based on BMI: underweight (<18.0 kg/m虏), normal (18.0 to <25.0 kg/m虏), and overweight (鈮�25.0 kg/m虏). For girls: 17 are underweight, 113 are normal, and 13 are overweight. For boys: 7 are underweight, 115 are normal, and 21 are overweight.
03

Selecting the Test

To assess differences in categorical distribution of weight status between boys and girls, we should use a Chi-square test for independence. This test evaluates if there is a significant association between two categorical variables鈥攊n this case, gender and weight status.
04

Setting Up Hypotheses

Formulate the null hypothesis as: 'There is no significant difference in the distribution of weight status between boys and girls.' The alternative hypothesis would be: 'There is a significant difference in the distribution of weight status between boys and girls.'
05

Executing the Test

Perform the Chi-square test by calculating expected frequencies for each category, then use these to determine the Chi-square statistic. Compare this statistic to a critical value from the Chi-square distribution table, or use software to find the p-value.
06

Interpretation of Results

If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis, indicating a significant difference in weight status distribution between boys and girls. If the p-value is greater, we do not reject the null hypothesis.

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Key Concepts

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

Body Mass Index (BMI)
Body Mass Index, commonly known as BMI, is a simple calculation used to assess whether a person's weight is appropriate for their height. It is calculated using the formula:\[BMI = \frac{\text{weight in kilograms}}{(\text{height in meters})^2}\] BMI is widely used because it's easy to measure and provides quick insights into whether someone falls into categories like underweight, normal weight, or overweight. These categories help indicate potential health risks related to weight.
  • Underweight: BMI is less than 18.0 kg/m虏.
  • Normal: BMI is between 18.0 and 24.9 kg/m虏.
  • Overweight: BMI is 25.0 kg/m虏 or higher.
For example, in the study, students were classified based on their BMI, allowing researchers to analyze weight categories effectively. While BMI is useful, it's important to remember that it does not directly measure body fat or account for muscle mass, which can affect interpretation.
gender differences
In many areas of health and social science, researchers are interested in exploring gender differences. This is because men and women can have different physiological and psychological characteristics that impact health differently. In the context of BMI and weight perception, gender differences may manifest in how males and females perceive their weight or respond to weight categories. For instance, societal norms and expectations often influence how body image and weight are perceived across genders. This is crucial when evaluating BMI data as it may affect both the reporting and interpretation of weight-related information. In the exercise, researchers used BMI data from boys and girls to see if there was a distinct pattern or discrepancy between the genders in terms of weight classification. Recognizing these differences can help in designing better health interventions and policies that cater specifically to the needs of different genders.
hypothesis testing
Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It begins with formulating two hypotheses:
  • Null Hypothesis (\(H_0\)): This is a statement of no effect or no difference. In our exercise, it states there is no significant difference in weight status between boys and girls.
  • Alternative Hypothesis (\(H_a\)): This suggests a significant effect or difference exists. Here, it posits a significant difference in weight status between genders.
After formulating these hypotheses, a statistical test, like the Chi-square test for independence, is conducted. This test assesses whether there is an association between two categorical variables (e.g., gender and weight status).During the test, expected frequencies are calculated under the assumption that the null hypothesis is true. The Chi-square statistic is then calculated using these expected values and the observed data. The result is compared to a critical value from the Chi-square distribution or a p-value is computed.
  • If the p-value is less than a pre-determined significance level (commonly 0.05), the null hypothesis is rejected, suggesting a significant difference exists.
  • If the p-value is greater, the null hypothesis is not rejected, implying no significant difference was detected.
Hypothesis testing is a powerful tool, helping convert data into actionable insights by quantitatively underscoring distinctions or similarities within a study.

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