Problem 2 - Algebra 2 and Trigonometry

Question

If you keep a record of the temperature in degrees Fahrenheit and in degrees Celsius for a month, what would you expect the correlation coefficient to be? Justify your answer.

Step-by-Step Solution

Answer

The correlation is 1, indicating a perfect linear relationship.

Step 1: Understanding Correlation Coefficient

The correlation coefficient measures the strength and direction of a linear relationship between two variables.

Step 2: Relationship Between Fahrenheit and Celsius

The relationship between temperatures in degrees Fahrenheit (6F) and degrees Celsius (6C) is linear, following the formula: 6F = 6C imes 9/5 + 32.

Step 3: Computing Correlation with Linear Transformation

The formula for a perfect linear transformation is y = ax + b, where 'a' and 'b' are constants. The formula 6F = 6C imes 9/5 + 32 fits this structure perfectly.

Step 4: Expecting Correlation Coefficient Value

When one variable is a perfect linear transformation of the other, as is the case here, the correlation coefficient is 1. This represents a perfect positive linear relationship.

Key Concepts

Linear Relationship

A linear relationship describes a straight-line connection between two variables. Imagine drawing a line on a graph; if a line passes through data points, it indicates a linear relationship. The correlation coefficient quantifies the strength and direction of this relationship. A value of +1 indicates a perfect positive linear relationship, meaning as one variable increases, the other does too, exactly proportionally.
For example, if temperatures in Celsius and Fahrenheit were plotted, you'd see they lie perfectly on a line. This is because the formula connecting them is linear, like a neatly drawn line that perfectly aligns with observed data points.

Fahrenheit and Celsius

The Fahrenheit and Celsius scales are two ways to measure temperature. Understanding them helps in interpreting their relationship.

• Fahrenheit (°F) is a temperature scale with water freezing at 32 °F and boiling at 212 °F.
• Celsius (°C) is another scale where water freezes at 0 °C and boils at 100 °C.

When you compare these scales, you're often converting from one to the other. That’s why understanding the formula connecting them is crucial; it allows seamless temperature conversion either way.

Temperature Conversion

Temperature conversion allows you to switch values from Celsius to Fahrenheit and vice versa. The mathematical formula used is: \[ F = C \times \frac{9}{5} + 32 \]This formula shows how Celsius (C) can be converted to Fahrenheit (F). The conversion is consistent because it uses a linear equation. The constant \(\frac{9}{5}\) represents the slope or rate of change, while 32 is the adjustment or intercept. Whenever you work through conversions, follow the formula to ensure accurate results.
This equation ensures every degree change in Celsius scales accurately to Fahrenheit.

Linear Transformation

A linear transformation is a type of function that maps input values to output values in a straight-line manner. In our context, we see it through temperature conversion. The formula \(y = ax + b\) is key to understanding linear transformations.
In the temperature formula:

• \(a = \frac{9}{5}\), which is the scaling factor transforming Celsius to match the degree span of Fahrenheit.
• \(b = 32\), the offset aligning the freezing points of the two scales.

Every transformation, whether in temperature scales or other mathematical contexts, follows this principle of consistent addition and multiplication, anchoring the concept of linearity.