Math  /  Data & Statistics

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, xx, and velocity is the dependent variable. (Round your numerical answers to two decimal places.) y^=\hat{y}=\square (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.) \qquad km/s\mathrm{km} / \mathrm{s}

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 y^=mx+b\hat{y} = mx + b, where mm is the slope and bb is the y-intercept.
3. The data points provided are: (51.8,4560),(12.2,1184),(27.1,1736),(46.2,3807),(58.2,5168),(46.2,3807),(29.1,1714)(51.8, 4560), (12.2, 1184), (27.1, 1736), (46.2, 3807), (58.2, 5168), (46.2, 3807), (29.1, 1714).

STEP 2

1. Calculate the slope mm of the regression line.
2. Calculate the y-intercept bb 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 (xx) and velocities (yy).
xˉ=51.8+12.2+27.1+46.2+58.2+46.2+29.17\bar{x} = \frac{51.8 + 12.2 + 27.1 + 46.2 + 58.2 + 46.2 + 29.1}{7}
yˉ=4560+1184+1736+3807+5168+3807+17147\bar{y} = \frac{4560 + 1184 + 1736 + 3807 + 5168 + 3807 + 1714}{7}

STEP 4

Calculate the slope mm using the formula:
m=(xixˉ)(yiyˉ)(xixˉ)2m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}

STEP 5

Calculate the y-intercept bb using the formula:
b=yˉmxˉb = \bar{y} - m\bar{x}

STEP 6

Write the equation of the linear regression line:
y^=mx+b\hat{y} = mx + b

STEP 7

Substitute x=130x = 130 into the regression equation to estimate the velocity:
y^=m(130)+b\hat{y} = m(130) + b
Round the result to the nearest whole number.
The equation of the linear regression line is:
y^=mx+b\hat{y} = mx + b
The estimated velocity for a galaxy 130 Mpc from Earth is:
km/s\mathrm{km/s}

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