Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Obtain the least squares regression line and the correlation coefficient. Make \(a\) statement about the correlation. Gestation Period and Life Span of Animals $$\begin{array}{|l|c|c|} \text { Animal } & \begin{array}{c} \text { Average Gestation } \\ \text { or Incubation Period, } \\ \boldsymbol{x} \text { (in days) } \end{array} & \begin{array}{c} \text { Record Life } \\ \text { Span, } \boldsymbol{y} \\ \text { (in years) } \end{array} \\ \hline \text { Cat } & 63 & 26 \\ \text { Dog } & 63 & 24 \\ \text { Duck } & 28 & 15 \\ \text { Elephant } & 624 & 71 \\ \text { Goat } & 151 & 17 \\ \text { Guinea pig } & 68 & 6 \\ \text { Hippopotamus } & 240 & 49 \\ \text { Horse } & 336 & 50 \\ \text { Lion } & 108 & 29 \\ \text { Parakeet } & 18 & 12 \\ \text { Pig } & 115 & 22 \\ \text { Rabbit } & 31 & 15 \\ \text { Sheep } & 151 & 16 \end{array}$$

Step-by-Step Solution

Answer

The regression line is \( y = 0.097x + 11.42 \) and the correlation coefficient \( r \approx 0.92 \), indicating a strong positive correlation.

Step 1: Calculate Means

First, we need to calculate the means of the gestation period (\( \bar{x} \)) and the life span (\( \bar{y} \)) for all animals. To do this, we sum all the values of \( x \) and \( y \) and divide by the total number of animals (13).

Step 2: Compute Sums for Least Squares

Calculate the sums needed for the slope and intercept of the regression line:\[S_{xy} = \sum{(x_i - \bar{x})(y_i - \bar{y})}\]\[S_{xx} = \sum{(x_i - \bar{x})^2}\]Use these to compute the slope \(b\) and intercept \(a\)\ of the regression line using: \( b = \frac{S_{xy}}{S_{xx}} \) and \( a = \bar{y} - b\bar{x} \).

Step 3: Calculate Regression Line

Using the slope \( b \) and intercept \( a \) computed in the previous step, the equation of the least squares regression line is given by:\[ y = bx + a \]

Step 4: Calculate Correlation Coefficient

To find the correlation coefficient \( r \), use the formula: \[ r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \]Where\[ S_{yy} = \sum{(y_i - \bar{y})^2} \]This gives the degree of linear relationship between gestation period and life span.

Step 5: Analyze Correlation

Based on the correlation coefficient \( r \), determine the strength and direction of the relationship. If \( r \) is close to 1 or -1, it indicates a strong linear relationship, whereas if \( r \) is close to 0, it indicates a weak relationship.

Key Concepts

Understanding the Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), measures the strength and direction of the linear relationship between two variables. In this context, it evaluates how the gestation period of an animal is related to its life span. A correlation coefficient can range between -1 and 1.
The closer \( r \) is to 1, the stronger the positive linear relationship, meaning as the gestation period increases, the life span also tends to increase. Conversely, a value of -1 indicates a strong negative linear relationship.

• If \( r \) is around 0, it suggests little to no linear relationship.
• An \( r \) around 0.5 suggests a moderate correlation.

In practical terms, the correlation coefficient helps us quantify and interpret the relationship so that we better understand how these biological variables interact.

Exploring Linear Relationships

A linear relationship implies that there's a proportional change between two variables. In the context of gestation period (\( x \)) and life span (\( y \)), a linear relationship means that changes in gestation period could predict changes in life span. To visualize this, we can plot the data on a scatter plot and draw a line that best fits.
This line, known as the least squares regression line, is used to predict life span based on gestation period. The equation of the line is given by:

• \( y = bx + a \)

Here, \( b \) represents the slope and \( a \) is the y-intercept. The slope \( b \) tells us how much \( y \) is expected to increase for each 1 unit increase in \( x \). If \( b \) is positive, it denotes a positive relationship and vice versa.

The Importance of Gestation Period

The gestation period is the duration an animal spends developing before birth. It can vary significantly between animals, ranging from a mere 18 days for a parakeet to a whopping 624 days for an elephant. It plays a crucial role in understanding some aspects of animal biology.
In studies like this, it's compared to other variables such as life span to uncover potential connections. Generally, animals with longer gestation periods may develop more complex anatomical features, possibly influencing their overall longevity.

• Understanding gestation helps in conservation efforts.
• It gives insights into evolutionary traits.

When evaluated alongside life span, we can infer patterns or trends that can be crucial for biological research and animal care strategies.

Interpreting Life Span Analysis

Life span analysis involves studying the life expectancy of different species, providing insights into the factors that influence longevity. In this context, it seeks to correlate life spans with gestation periods, highlighting biological complexities.
Our goal is to see if a systematic relationship exists. For instance:

• Do animals with longer gestation have longer life spans?
• How significant is this gestation period in predicting life span?

While a longer life span can imply better survival strategies or superior biological mechanisms, interpreting the data helps us understand evolutionary advantages. This analysis can lead to more tailored conservation programs and enhanced understanding of animal aging processes.