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A study conducted at the University of Denver investigated whether babies take longer to learn to crawl in cold months, when they are often bundled in clothes that restrict their movement, than in warmer months. \({ }^{6}\) Infants born during the study year were split into twelve groups, one for each birth month. We consider the average crawling age of babies in each group against the average temperature when the babies are six months old (that's when babies often begin trying to crawl). Temperature is measured in degrees Fahrenheit ( \({ }^{\circ} \mathrm{F}\) ) and age is measured in weeks. (a) Describe the relationship between temperature and crawling age. (b) How would the relationship change if temperature was measured in degrees Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and age was measured in months? (c) The correlation between temperature in \({ }^{\circ} \mathrm{F}\) and age in weeks was \(r=-0.70\). If we converted the temperature to \({ }^{\circ} \mathrm{C}\) and age to months, what would the correlation be?

Step by step solution

01

Understand the Context

We are investigating whether the average crawling age of babies relates to the average temperature (when they are six months old), measured in different birth month groups. Temperature influences crawling age, which we analyze using correlation.

02

Relationship Description

In part (a), the correlation coefficient given between temperature in Fahrenheit and crawling age in weeks is \( r = -0.70 \). A negative correlation indicates that as temperature increases, the crawling age decreases, i.e., in warmer months, babies tend to crawl earlier.

03

Conversion of Units

For part (b), to convert temperature from Fahrenheit to Celsius, use the formula: \( C = \frac{5}{9}(F - 32) \). Also, convert age from weeks to months: since 1 month ≈ 4.345 weeks, divide the age in weeks by 4.345.

04

Correlation Under Unit Conversion

For part (c), we use the idea that correlation is unitless and remains unchanged by linear transformations. Unit conversions for temperature and time are linear (e.g., \( C \) is a linear function of \( F \), and months is a linear function of weeks). Thus, the correlation value remains \( r = -0.70 \) despite these conversions.

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Key Concepts

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

Unit Conversion

Unit conversion is a process where measurements are transformed from one set of units to another. In the context of the exercise, this involves changing temperature from Fahrenheit to Celsius, and age from weeks to months.
To convert Fahrenheit to Celsius, we use the formula: \[ C = \frac{5}{9}(F - 32) \]This formula derives from the need to offset and scale the interval difference between the two temperature scales.
On the other hand, when converting weeks to months, since 1 month is approximately equal to 4.345 weeks, you convert the age by dividing the number of weeks by 4.345.

Unit conversion is crucial as it allows us to describe and interpret data in different contexts without changing the inherent relationship between the variables.

Negative Correlation

A negative correlation implies that as one variable increases, the other decreases. In the scenario provided, the correlation coefficient of \( r = -0.70 \) indicates a strong negative relationship between temperature and the crawling age of babies. As the temperature rises, babies tend to begin crawling earlier.
This phenomenon can be explained by noting that in warmer months, infants may wear less restrictive clothing, which could facilitate earlier crawling.

Understanding the sign and magnitude of a correlation coefficient helps interpret data relationships effectively, laying the groundwork for predicting and explaining patterns observed in datasets.

Data Interpretation

Data interpretation is the process of making sense of numerical data that has been collected, analyzed, and presented. In the context of this exercise, we interpret the relationship between temperature and crawling age through the lens of correlation analysis.

• The given correlation coefficient \( r = -0.70 \) suggests a significant relationship.
• Temperature data measured at when babies are six months old plays a critical role in establishing this relationship.

Interpreting such data requires understanding that correlation does not imply causation; thus, other factors might contribute to this observed pattern.
This careful analysis allows researchers to draw meaningful conclusions from complex statistical information.

Statistical Relationship

A statistical relationship conveys how two variables relate to each other through statistical measures, like correlation. In this exercise, we focus on how temperature and crawling age correlate. Correlation, being a statistical measure, quantitatively expresses the strength and direction of a linear relationship between two variables.
It is important to note that the statistical relationship remains unchanged under linear unit transformation (e.g., altering from Fahrenheit to Celsius). This property simplifies comparing relationships across various scales.

• Temperature is the independent variable, affecting crawling age, the dependent variable.
• The correlation coefficient value remains constant despite unit conversions, as it is a dimensionless measure.

Recognizing these relationships aids in predicting behavior, thus informing further research or practical applications.