Question

ALGEBRA HELP- Does anyone with patience know to do the “Least Squares Linear Regression Method”? I watched a YouTube video on how to do it but I don’t know how to apply it here. Sorry this one is long…and I really want to understand all of it, but if you can help with even one part- please do.
The table lists the distances (in megaparsecs; 1 megaparsec= 3.085 × 10^24 cm, and 1 megaparsec= 3.26 million light-years) and velocities (in kms per second) of four galaxies moving rapidly away from Earth.
(I’ll put the table in the comments)
(a) Plot the data using distances for the x-values

Answer 1

(cont.) and velocities for the y-values. What type of relationship seems to hold between the data?

(b) Find a linear equation in the form y=mx that models these data using the points (520, 40,000) and (0, 0). Graph your equation with the data on the same coordinate axes.
(c) The galaxy Hydra has a velocity of 60,000 km per sec. How far away is it according to the model in part (b)?

(d) The value of m is called the Hubble constant. The Hubble constant can be used to estimate the age of the universe A (in years) using the formula A= (9.5)(10^11)/m

(e) Astronomers currently place the value of the Hubble constant between 50 and 100. What is the range for the age of the universe A?

Answer 2

No one brave enough to tackle this one? lol

Answer 3

Did you want to go through all the parts?

Answer 4

This actually doesn't look too bad. I'm assuming that your class wants you to do this by hand?

Answer 5

Yes.
I know it should be easy which makes me feel stupid for asking...

Answer 6

No, no worries. It's only easy because I've had to pull my hair out on these in the past.

Answer 7

Have you already gone ahead and graphed the data for part a)?

Answer 8

Yes. I dont see the relation though

Answer 9

OK, I'm going to go ahead and use technology on my side to see the graph. Hold on one moment.

Answer 10

Alright, thank you!

Answer 11

OK, here's what I see:

Answer 12

Your graph is much more proportional than mine,haha, and I see the curve now

Answer 13

Well, I cheated with technology. :) Alright, so we see the curve now.

Answer 14

It's actually very straight, so a line should do a good approximation.

Answer 15

So its linear? is that the relationship?

Answer 16

Yep, one of the few words in math that actually makes some sense. :)

Answer 17

Alright, so now for part b).... whoa, hang on... this is a lot simpler than least square regression... you won't need that level of complexity...

Answer 18

So how do we make this into a linear equation? Take a random pair of coordinates?

Answer 19

Yeah I don't know. Our teacher had as watch a video about it before answering this question.

Answer 20

Yeah, probably not needed for this one at least... your intuition into taking a random pair of coordinates is the best move. Actually the question says to use the points (520,42000) and (0,0)

Answer 21

So, they've chosen the random points for us.

Answer 22

So, part b) wants us to write something for y = mx. The main challenge right now is to find m, the slope. Do you have any ideas based on the points they gave us?

Answer 23

So, 420000=m(520)
Because you're just subracting the first coordinantes by the second pair which are zeros

Answer 24

OK, looks good

Answer 25

So, m=807.692

Answer 26

I think you mean 80.769, I believe it was 42,000 vs. 420,000, but I might be mistaken.

Answer 27

Actually, wait... The points were (0,0), and (520,40,000) typo on my part. Sorry.

Answer 28

Oops it was 40,000. Guess we dragged the 2 from 520 by mistake

Answer 29

Yep, that happened. :) So, I'm getting 76.923

Answer 30

yeah haha. So 40000=m(520)
So m=76.923

Answer 31

Yay!!

Answer 32

So our equation is y = 76.923 x, where x is the distance and y is the velocity.

Answer 33

I'll go ahead and plot this over my previous graph, one second.

Answer 34

Here we go

Answer 35

So for part C, I plug in the velocity and y=(76.923)(60,000) ?

Answer 36

Careful... the velocity goes into the y, not the x.

Answer 37

Almost linear, do I needto change answer A?

Answer 38

Actually it would be a good idea to write your equation as: v = 76.923 d to keep that clear.

Answer 39

No need to change your answer. In the real world, there is almost NOTHING that is perfectly described by an equation. It is "close enough" so we would say it is linear.

Answer 40

Actually, this equation does an amazing job of fitting the data, and any scientist would be very happy with this. :)

Answer 41

Okay, then linear it is. And galaxy Hydra is 780 (megaparsec?) away?

Answer 42

Let me check.

Answer 43

Yep, I agree.

Answer 44

Now to estimate the age of the universe with our slope.

Answer 45

Does the m stand for megaparsec?

Answer 46

I don't think so, mega is usual a capital M in science. So I'm pretty sure they are referring to m as in the slope of the line. Also, is the formula written like this? \[\frac{9.5 \times 10^{11}}{m}\]

Answer 47

yes

Answer 48

OK, I think all they want is to just plug in our m = 76..... into the equation.

Answer 49

Do I have to simplify it at all?

Answer 50

Sure, it will be easier to understand that way... you'll need a powerful calculator though. Your computer calculator can probably handle it or ask google. :)

Answer 51

I'm getting a really big number...

Answer 52

Me too :(

Answer 53

I'm using mathway, what are you using?

Answer 54

Just searching directly in Google the following
9.5*10^11 / 76.923

gives me this
12350012350

Answer 55

Looks like it has a built in calculator, this agrees with my other one, it just dropped a few decimals.

Answer 56

I got 12348888600.026 so... 1.23x10^11 ? Does that sound reasonable?

Answer 57

Sure, our model isn't perfect anyway, so that's a good estimate.

Answer 58

So I have no lead for what part E wants me to do

Answer 59

Just real quick I think you want 1.23x10^10

Answer 60

Oh okay. Sounds good. Good to double check, ty

Answer 61

OK, so E is just a really complicated way of them telling you to do part D again. Except this time use m = 50 and see what you get. Then use m = 100 and see what you get. Then you'll see that this is a range of possible ages for the universe. m = 50 gives the maximum age and m = 100 gives the minimum age.

Answer 62

Ohhh okay!! Thank you so much!!!

Answer 63

Your welcome! Not so bad hopefully! :)

Answer 64

Not as bad as I thought it would be at all XD guess I'm just afraid of big numbers

Answer 65

They are scary, but we can still deal with them. Just takes more paper and typing. :) Good work! Take care!

Answer 66

If you want me to check the other two answers you get, just let me know. In fact, I'll post mine below so you can double check.

Answer 67

m = 100 -> 9,500,000,000 = 9.5 x 10^9
m = 50 -> 19,000,000,000 = 1.9 x 10^10

Answer 68

Yes! B) Yay team! Have a good *enter time of the day here*

Answer 69

lol Sounds good, you too!