Problem 71 - A Graphical Approach to Precalculus with Limits

Question

In the late 1920 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 below. $$\begin{array}{|l|c|c|} \hline \multicolumn{1}{|c|}\text { Galaxy } & \text { Distance } & \text { Velocity } \\ \hline \text { Virgo } & 50 & 990 \\ \text { Ursa Minor } & 650 & 9,300 \\ \text { Corona Borealis } & 950 & 15,000 \\ \text { Bootes } & 1,700 & 25,000 \end{array}$$ (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

Regression line: \( y = 0.0107x + 702.12 \); Hydra distance: 1,098 light-years.

Step 1: Calculate the Means

To find the equation of the least-squares regression line, first calculate the means of the given distance (\(y\)) and velocity (\(x\)) data.\[ \bar{x} = \frac{(990 + 9300 + 15000 + 25000)}{4} = 12572.5 \]\[ \bar{y} = \frac{(50 + 650 + 950 + 1700)}{4} = 837.5 \]

Step 2: Calculate the Slopes

Next, calculate the slope \( m \) of the regression line using:\[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \]Calculate \( \sum{(x_i - \bar{x})(y_i - \bar{y})} \) and \( \sum{(x_i - \bar{x})^2} \):\[ \sum{(x_i - \bar{x})(y_i - \bar{y})} = (990-12572.5)(50-837.5) + (9300-12572.5)(650-837.5) + (15000-12572.5)(950-837.5) + (25000-12572.5)(1700-837.5) = 1540875 \]\[ \sum{(x_i - \bar{x})^2} = (990-12572.5)^2 + (9300-12572.5)^2 + (15000-12572.5)^2 + (25000-12572.5)^2 = 144492725 \]Now calculate \( m \):\[ m = \frac{1540875}{144492725} \approx 0.0107 \]

Step 3: Determine the Intercept

Calculate the y-intercept \( b \) using the formula:\[ b = \bar{y} - m \bar{x} \]Substitute the calculated values:\[ b = 837.5 - 0.0107 \times 12572.5 \approx 702.12 \]

Step 4: Write the Regression Equation

Now, write the equation of the least-squares regression line:\[ y = mx + b \]With the determined slope and intercept:\[ y = 0.0107x + 702.12 \]

Step 5: Estimate Distance for Hydra

To estimate the distance for the galaxy Hydra with a velocity of 37,000 miles per second, use the same regression equation:\[ y = 0.0107 \times 37000 + 702.12 \]Calculate:\[ y = 395.9 + 702.12 = 1098.02 \]The estimated distance is approximately 1,098 light-years.

Key Concepts

Hubble's Law

Hubble's Law is a significant discovery in the field of astronomy. Introducing the relationship between a galaxy's distance and its velocity, this law lays the groundwork for how we perceive the universe's expansion. Simply put, Hubble's Law states that the further away a galaxy is, the faster it appears to be moving away from us. This observation led to the idea that the universe is constantly expanding. In mathematical terms, Hubble's Law can be expressed as \( v = H_0 \times d \), where \( v \) represents the velocity of the galaxy moving away from the Earth, \( H_0 \) is the Hubble constant, and \( d \) is the distance from the Earth.
This law forms the basis for understanding cosmology and the movement and behavior of galaxies in the cosmos.

• It explains the redshift observed in the light from distant galaxies.
• It supports the Big Bang theory, suggesting that the universe started from a single point.
• It helps astronomers estimate the size and age of the universe.

Understanding Hubble's Law is crucial for anyone studying astronomy or engaged in astrophysical research.

Slope Calculation

Calculating the slope is a critical step in determining the relationship between two variables, such as distance and velocity in our galaxy data. The slope of a line in a regression model represents the rate at which one variable changes with respect to another. Here, we focus on how velocity affects distance.
The slope \( m \) can be calculated using the formula: \[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \] where:

• \(x_i\) and \(y_i\) are the individual data points of velocity and distance, respectively.
• \(\bar{x}\) and \(\bar{y}\) are the mean values of the velocity and distance.

The numerator \( \sum{(x_i - \bar{x})(y_i - \bar{y})} \) gives the covariance, showing how changes in one variable relate to changes in the other. The denominator \( \sum{(x_i - \bar{x})^2} \) is the variance of the velocity data. Divide these sums to find \( m \), which signifies how much the distance changes for each unit change in velocity. In this example, it was approximately \(0.0107\). Having a positive slope indicates a direct proportional relationship between velocity and distance.

Intercept Calculation

The intercept calculation gives us another crucial component of our regression line. The intercept, represented as \( b \), tells us the expected value of the dependent variable (distance, in this case) when the independent variable (velocity) equals zero.
The formula used is: \[ b = \bar{y} - m \bar{x} \] where:

• \(\bar{y}\) is the mean of the distance values.
• \(\bar{x}\) is the mean of the velocity values.
• \(m\) is the slope calculated previously.

By substituting the known values, we can calculate \( b \), which gives insight into one part of the initial value setting in the described statistical relationship. For our Hubble's galaxy data, we found \( b \approx 702.12 \). This value is important because it sets the baseline from which linear predictions can be made under this model.

Statistical Modeling

Statistical modeling is a powerful tool for analyzing and making predictions based on data. In this context, the least-squares regression line is a fundamental type of statistical model that helps to describe the linear relationship between two quantitative variables—here, distance and velocity of galaxies.
This model minimizes the discrepancy between observed values and the values predicted by the model. By calculating means, slopes, and intercepts, we construct a line that best fits the data points available.
The general form of a linear regression equation is: \[ y = mx + b \] where:

• \( y \) is the predicted value for the dependent variable.
• \( m \) is the slope of the line.
• \( x \) is an independent variable.
• \( b \) is the y-intercept.

Once formulated, this equation can be used to make predictions, such as estimating the distance of the Hydra galaxy given its velocity. These predictions are valuable in many scientific domains, including physical sciences and economics.