The Variables
The data set has the variables Velocity and Distance. Here is some info on these:
Velocity
Velocity (Speed with a sign) is measured in km/sec. How can one measure the speed with which a galaxy moves relative to earth? This is done using the Doppler Effect:
For more on the Doppler Effect go to Doppler Effect.
Distance
The unit of distance in our dataset is one Megaparsec, or 1 million parsecs. A parsec is equal to 3.262 light years, or 19.17 billion miles. Here are some astronomical distances for illustration:
• Earth to Moon: 240000 miles (or 1.3 light seconds)
• Earth to Sun: 92 million miles (or 8.2 light minutes)
• Earth to nearest Solar System (Alpha Centauri): 8.37 billion miles (or 4.365 light years or 1.338 parsec)
• Earth to nearest galaxy (Andromeda): 48000 million million miles (or 2.5 million light years or 740 000 parsec or 0.7 megaparsec)
How does one measure the distance of a galaxy (or a star)? It is done using a method called parallax:
For more on parallax go here
Data Analysis
A scatterplot of Velocity by Distance shows a strong positive relationship (r=0.790, p-value = 0.000)
The least squares regression equation is Velocity = - 40.8 + 454 Distance.
Stat > Regression > Regression, Response= Velocity, Predictors= Distance, Graphs > Normal Plot and Residuals vs. Fits Plot
Assumptions:
• the residual vs. fits plot looks fine, so a linear model is good
• the normal plot is fine, no problems with the normal assumption
• the residual vs. fits plot shows no problems with the equal variance assumption
Tests:
Distance:
1) a=0.05
2) H0: b1 = 0 (no relationship between distance and velocity)
3) Ha: b1 ≠ 0 (some relationship between distance and velocity)
4) p=0.000
5) p<a, so we reject the null hypothesis, there is a statistically significant relationship between velocity and distance. (Note: this is exactly the same test as the Pearson correlation coefficient test!)
Constant:
1) a=0.05
2) H0: b0 = 0 (intercept is zero)
3) Ha: b0 ≠ 0 (intercept is not zero)
4) p=0.630
5) p>a, so we fail to reject the null hypothesis, the constant is stat. consistent with 0 (at the sample size of the dataset)
Because of this we might refit a no-intercept model:
Stat > Regression > Regression, Response= Velocity, Predictors= Distance, Graphs > Normal Plot and Residuals vs. Fits Plot, Options: uncheck Fit Intercept
Velocity = 424 Distance
This number 424 is called Hubble's constant
What it all means
What are the consequences of all this for our understanding of the universe?
Some more stuff
The value of the Hubble Constant initially obtained by Edwin Hubble was
around 500 km/s/Mpc, and has since been radically revised because initial
assumptions about stars yielded underestimated distances.
For the past three decades, there have been two major lines of investigation
into the Hubble Constant. One team, associated with Allan Sandage of the
Carnegie Institutions, has derived a value for Ho around 50 km/s/Mpc. The
other team, associated with Gerard DeVaucouleurs of the University of
Texas, has obtained values that indicate Ho to be around 100 km/s/Mpc. A
long-term, key program for HST is to refine the value of the Hubble
Constant.
Hubble's original data are now known to be flawed. Galaxies near our own Milky Way move around each other forming both a local "cluster" of galaxies and a local
"supercluster" of galaxy clusters. The movement of nearby galaxies relative to
the Milky Way is affected by their motions within these clusters, and does not
represent only the expansion of the universe. Likewise, the measurement of distance
over such a great expanse has been a subject of continuing research. We now have
estimates of the distances to nearby galaxies that are thought to be much more accurate
than Hubble's original attempts. Nevertheless, the original data have historical
importance. And, of course, we now know that while the estimated slope of Hubble's
regression was probably wrong, the conclusion of the hypothesis test, that there is
a positive relationship between distance and velocity, has been resoundingly confirmed;
we see it wherever we look in the universe.
Using the much more recent data also in the datafile we find the equation velocity(km/s) = 73.2 Distance(Mpc), or a Hubble constant of 73.2
The Hubble Constant also allows us to approximately calculate the age of the universe. The age is approximately the inverse of the Hubble Constant. The hubble constant is 73.238km/s/Mpc. The inverse of this is, .01365s/Mpc/km, converting from Mpc to km and cancelling the two out, gives you 409120663320968797 seconds, or 12.973billion years