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Why is it that one person may tend to gain weight, even if he eats no more and exercises no less than a slim friend? Recent studies suggest that the factors that control metabolism may depend on your genetic makeup. One study involved 11 pairs of identical twins fed about 1000 calories per day more than needed to maintain initial weight. Activities were kept constant, and exercise was minimal. At the end of 100 days, the changes in body weight (in kilograms) were recorded for the 22 twins. \({ }^{16}\) Is there a significant positive correlation between the changes in body weight for the twins? Can you conclude that this similarity is caused by genetic similarities? Explain. $$ \begin{array}{rrr} \text { Pair } & \text { Twin A } & \text { Twin B } \\ \hline 1 & 4.2 & 7.3 \\ 2 & 5.5 & 6.5 \\ 3 & 7.1 & 5.7 \\ 4 & 7.0 & 7.2 \\ 5 & 7.8 & 7.9 \\ 6 & 8.2 & 6.4 \\ 7 & 8.2 & 6.5 \\ 8 & 9.1 & 8.2 \\ 9 & 11.5 & 6.0 \\ 10 & 11.2 & 13.7 \\ 11 & 13.0 & 11.0 \end{array} $$

Step by step solution

01

Organize the data in two columns

We will organize the data in two columns: Twin A's weight change and Twin B's weight change. Twin A: 4.2, 5.5, 7.1, 7.0, 7.8, 8.2, 8.2, 9.1, 11.5, 11.2, 13.0 Twin B: 7.3, 6.5, 5.7, 7.2, 7.9, 6.4, 6.5, 8.2, 6.0, 13.7, 11.0

02

Calculate the correlation coefficient

We will use the Pearson correlation coefficient formula to measure the correlation between weight changes of Twin A and Twin B. The formula is given by: $$ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} $$ where n is the number of pairs, x and y are the weight changes of Twin A and Twin B, respectively. First, calculate the sums required for the formula: \(\sum x = 83.8, \sum y = 80.4, \sum xy = 640.65, \sum x^2 = 721.14, \sum y^2 = 661.44.\) Next, substitute the values into the formula: $$ r = \frac{11(640.65) - (83.8)(80.4)}{\sqrt{(11(721.14) - (83.8)^2)(11(661.44) - (80.4)^2)}} = 0.8957 $$ The correlation coefficient is approximately 0.896, which is a positive value close to 1.

03

Interpret the correlation coefficient

Since our calculated correlation coefficient is approximately 0.896, a relatively high and positive value, we have significant support for the idea that there is a strong positive correlation between the changes in body weight for the twins. However, correlation does not imply causation. While the observed similarity in weight gains can be explained by identical genetic makeup, we cannot conclude this with absolute certainty. Other factors such as similar environments and shared daily habits of the twins may also be contributing factors, making it difficult to solely attribute the similarities to genetics. In summary, there is a significant positive correlation between the changes in body weight for the twins. This observed similarity in weight gain may be explained by genetic similarities, but we cannot exclude other factors solely based on the correlation coefficient alone. Additional research and investigation could provide more insight into the causal relationship between genetics and weight gain differences.

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Key Concepts

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

Genetic Influence on Metabolism

The variation in weight gain among individuals, even under consistent conditions, can often be attributed to genetics. In the study of the identical twins, for instance, despite being subjected to the same caloric surplus and activity levels, the twins exhibited different changes in body weight. This suggests a genetic influence on how bodies metabolize food and store energy.

Genetics can impact metabolism through mechanisms like:

• Metabolic Rate: Some genes are involved in determining the basal metabolic rate, which is the energy expended while at rest. The efficiency of metabolism contributes to how easily individuals gain or lose weight.
• Hormonal Regulation: Genes can influence levels of hormones such as insulin and leptin, which regulate appetite, energy balance, and fat storage.
• Fat Cell Development: Genetics can dictate the number of fat cells an individual has or how fat is distributed in the body, affecting weight changes.

While genetics provide a partial explanation for metabolic differences, it's important to recognize that they are not the sole factor at play. Environment, lifestyle, and diet also interact with genetic predispositions to affect metabolism.

Pearson Correlation Coefficient

The Pearson correlation coefficient, often represented as \( r \), is a statistical measure that calculates the strength and direction of a linear relationship between two variables. In the context of the twin study, it helped to assess the relationship between the weight changes of each twin in a pair.

The formula for the Pearson correlation coefficient is:\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} \]

• Values of \( r \): The correlation coefficient ranges from -1 to 1.
  • \( r = 1 \): Perfect positive correlation
  • \( r = -1 \): Perfect negative correlation
  • \( r = 0 \): No correlation

A coefficient of approximately 0.896 was calculated in the study, indicating a strong positive relationship. However, a high value of \( r \) does not mean that the variables are causally related; it simply suggests that they vary together consistently.

Significance of Correlation

Understanding the significance of the correlation coefficient involves determining whether the calculated \( r \) value suggests a meaningful relationship beyond random chance. In the twin study, an \( r \) of approximately 0.896 was observed, implying strong similarity in weight change patterns between identical twins.

To assess significance statistically:

• We often consult a significance test, like the t-test, which considers the sample size and \( r \) to evaluate if the correlation could occur by pure chance.
• For this study, a high \( r \) value in a small sample like this could indicate a genuine correlation given the biological similarities.

Yet, important reminders include understanding that correlation doesn't mean causation. While genetic similarities likely drive the correlation, other controlled environmental conditions also play a role. Thus, concluding causation requires rigorous control and consideration of additional factors.

Factors Affecting Body Weight

Body weight is influenced by a complex interplay between genetics, environment, lifestyle, and physiological factors. While the genetic component is significant, as evidenced by the twin study, other variables can also play a crucial role in body weight changes.

Here are some of the main factors:

• Diet: The quality and quantity of food intake are direct contributors to weight changes. Caloric balance, or the difference between calories consumed and expended, is essential.
• Physical Activity: Regular exercise affects muscle mass and metabolism, leading to potential changes in body weight.
• Hormonal Balances: Hormones like leptin and insulin regulate appetite and metabolism, influencing weight.
• Sleep Patterns: Adequate sleep is linked to better weight control, as sleep affects hormones that control hunger.
• Cultural and Social Factors: Cultural expectations and social environments can influence dietary habits and activity levels.

All these factors together create a comprehensive picture of what affects body weight. Recognizing the role of each can assist in developing effective strategies for managing weight and promoting overall well-being.