QuestionFollow the link Least Squares Line. This will direct you to a spreadsheet download that may be useful for checking your work for the exercise. Astronomer Edwin Hubble postulated a relationship between the distance between Earth and the velocity at which a galaxy appears to be traveling away from Earth. The following table shows observations of seven galaxies. Distance is measured in megaparsecs ( 1 Mpc is approximately 3,260 light-years), and velocity is measured in kilometers per second. \begin{tabular}{|c|c|} \hline Distance (Mpc) & Velocity (km/s) \\ \hline 51.8 & 4,560 \\ \hline 12.2 & 1,184 \\ \hline 27.1 & 1,736 \\ \hline 46.2 & 3,807 \\ \hline 58.2 & 5,168 \\ \hline 46.2 & 3,807 \\ \hline 29.1 & 1,714 \\ \hline \end{tabular} (a) Find the equation of linear regression line for the data where distance is the independent variable, , and velocity is the dependent variable. (Round your numerical answers to two decimal places.) (b) Using the equation from part (a), estimate the velocity (in kilometers per second) at which a galaxy 130 Mpc from Earth is traveling. (Round your answer to the nearest whole number.)
Studdy Solution
STEP 1
1. The relationship between distance and velocity is linear.
2. The equation of the linear regression line is of the form , where is the slope and is the y-intercept.
3. The data points provided are: .
STEP 2
1. Calculate the slope of the regression line.
2. Calculate the y-intercept of the regression line.
3. Write the equation of the linear regression line.
4. Use the regression equation to estimate the velocity for a galaxy 130 Mpc from Earth.
STEP 3
Calculate the mean of the distances () and velocities ().
STEP 4
Calculate the slope using the formula:
STEP 5
Calculate the y-intercept using the formula:
STEP 6
Write the equation of the linear regression line:
STEP 7
Substitute into the regression equation to estimate the velocity:
Round the result to the nearest whole number.
The equation of the linear regression line is:
The estimated velocity for a galaxy 130 Mpc from Earth is:
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