Problem 71 - A Graphical Approach to College Algebra

Question

In the late \(1920 \mathrm{s}\), the famous observational astronomer Edwin P. Hubble ( \(1889-1953\) ) determined the distances to several galaxies and the velocities at which they were receding from Earth. Four galaxies with their distances in light-years and velocities in miles per second are listed in the table at the top of the next column. $$\begin{array}{|l|c|c|}\quad\quad\text { Galaxy } & \text { Distance } & \text { Velocity } \\\\\hline \text { Virgo } & 50 & 990 \\\\\text { Ursa Minor } & 650 & 9,300 \\\\\text { Corona Borealis } & 950 & 15,000 \\\\\text { Bootes } & 1,700 & 25,000\end{array}$$ (a) Let \(x\) represent velocity and \(y\) represent distance. Find the equation of the least-squares regression line that models the data. (b) If the galaxy Hydra is receding at a speed of \(37,000\) miles per second, estimate its distance from Earth.

Step-by-Step Solution

Answer

The least-squares regression line is approximately \\(y = 0.067x + 390\\). For Hydra, the estimated distance is about \\2,879\\ light-years.

Step 1: Calculate the Slope (m)

First, identify the coordinates for each galaxy as \(x, y\), where \x\ is the velocity and \y\ is the distance:\\[(990, 50), (9300, 650), (15000, 950), (25000, 1700)\]\.Calculate the mean of the velocities \(\bar{x}\) and the mean of the distances \(\bar{y}\). Then, use these to calculate the slope \(m\) of the least-squares regression line using the formula: \[m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}\]\. Plug in the values and calculate.

Step 2: Calculate the Intercept (b)

Now that we have the slope \(m\), calculate the intercept \(b\) using the formula: \[b = \bar{y} - m\bar{x}\]\.Substitute the values of \(\bar{x}\), \(\bar{y}\), and \(m\) computed in the previous step to calculate \(b\).

Step 3: Write the Equation of the Line

With the slope \(m\) and intercept \(b\) known, the equation of the least-squares regression line is y = mx + b\.Put the values calculated for \(m\) and \(b\) into the equation.

Step 4: Estimate Distance for Hydra

To estimate the distance \(y\) for Hydra, use the regression line equation. Substitute \(x = 37000\) into the equation obtained in Step 3 and solve for \(y\).

Key Concepts

Regression Line Equation

In statistical analysis, the regression line equation is a powerful tool. It's often used when you want to show the relationship between two quantitative variables. In the context of Edwin Hubble's data, this equation helps us understand the connection between galaxy velocity and distance from Earth. The equation has the general form: \( y = mx + b \). Here:

• \(y\) is the dependent variable, in this case, the distance of the galaxy.
• \(x\) is the independent variable, which refers to the velocity at which a galaxy is moving away.
• \(m\) is the slope of the line.
  It shows how much \(y\) changes for a one-unit increase in \(x\).
• \(b\) is the y-intercept.
  It's the value of \(y\) when \(x=0\).

To find \(m\), you use a formula that involves the means of both \(x\) and \(y\), and sums of the products of differences from these means. Similarly, you calculate \(b\) once you know \(m\). This method, known as the least squares method, minimizes the distance between the observed points and the estimated line, providing the best possible linear representation of the data.

Velocity and Distance Correlation

Velocity and distance are two key variables that often display a relationship in astronomical studies. This relationship is particularly evident in Hubble's observations.

• Velocity: In Hubble's context, this is the speed at which a galaxy is moving away from Earth.
  The faster a galaxy moves, the further away it might be. This speed is measured in miles per second.
• Distance: This indicates how far the galaxy is from Earth, typically measured in light-years.
  Greater distances suggest more significant travel times for light, indicating an older view of the universe's history.

The correlation between velocity and distance is quite logical in Hubble's framework. If a galaxy is moving quickly, it is generally found to be farther away. This observation eventually led Hubble to formulate what is known as Hubble’s Law. This law implies a linear relationship between velocity and distance, suggesting that the universe is expanding, which was a revolutionary idea at the time.

Edwin Hubble Observational Data

Edwin Hubble was a pioneering astronomer whose work fundamentally changed our understanding of the universe. In the late 1920s, Hubble collected observational data that documented distances of galaxies and the speeds at which they were receding from Earth.

• The crucial element of Hubble's observations was that he found a consistent pattern:
  more distant galaxies moved away faster than closer ones.
• Hubble's data was gathered by measuring the redshift of light from galaxies, which indicates how far away they're receding.
• This data set provided the empirical foundation for Hubble's Law,
  which states that the farther a galaxy is, the faster it is moving away from us.

Hubble's findings were critical in the shift from a static view of the universe to one that is expanding. His pioneering work also laid the groundwork for the Big Bang theory, which describes the universe's origins as a rapidly expanding entity from an initial singularity.