Question

WILL AWARD MEDAL!
George has found a relationship between the number of trees in an orchard and the number of species of birds found in the orchard. The table below shows the data collected by George:


Number of trees
(x) 100 200 300 400 500 600 700 800 900
Number of birds
(y) 51 101 151 201 251 301 350 400 450


Part A: What would most likely be the number of birds in the orchard if there are 284 trees?

Part B: Predict a possible correlation coefficient for the data in the table and explain why you think your prediction is a good value for the data.

Answer 1

Part C: George says that if water sources are provided in the orchard, the bird population would increase. Is this an example of correlation or causation? Justify your answer.

Answer 2

@marissalovescats

Answer 3

@austinisawesome12

Answer 4

hi

Answer 5

let em see

Answer 6

@Clalgee

Answer 7

@wio

Answer 8

@imer

Answer 9

DO you know the regression line equation?

Answer 10

Answer for part A should be "143"

300 Trees = 151 Birds
200 Trees = 101 Birds

300-200=151-101
Difference: 100 Trees = 50 Birds
Twice as many trees as birds.

Therefore if we follow the pattern;
300-284=x
Difference: 16 Trees = 8 Birds

Answer 11

The regression line is \[y=\hat{\beta_0}+\hat{\beta_1}x \] where
\[\hat{\beta_0}=\bar{y}-\hat{\beta_1}\bar{x} \] and
\[\hat{\beta_1}=\large \frac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2} \]

Answer 12

Once you find \(\hat{\beta_0}\) and \(\hat{\beta_1}\), go back to the line equation \(y=\hat{\beta_0}+\hat{\beta_1}x\) and plug in \(x=284\)

Answer 13

It should be close though to 143 @imer said because the data is almost perfectly linear (as in question b), the correlation coefficient is veryyyy close to 1, and is probably around 0.999 or smtg like that)

Answer 14

You can probably just use software (R, SPSS, even Wolfram I think can do it) though to fit the line using your data, It's kinda long to do that by hand

Answer 15

@kirbykirby The question is not sophisticated as you can see the word "most likely" in part "A" and since the values are discrete, I am pretty sure the question does not demand knowledge about regression line.

Answer 16

Hm ya as I was answering myself, I think I realized it's not such a sophisticated question as you mentioned, although that realization came from question b) to me as it said to predict the value.. and certainly any program would just spit it out haha. Sorry for the confusion on my part :(

Answer 17

so for part b our predicted correlation coefficient value is 1?
and what about part C?

Answer 18

you guys have been an incredible help by the way

Answer 19

Probably correlation. Proving causation is actually quite difficult. Also, there is very little information in the question that says what George has done in the study to show this. But most likely, he did an observational study (you don't control the experiment yourself, meaning he can't control the number of birds visiting the orchard). Typically you can try to reach causation by doing an randomized experimental study. You can assign things to certain groups (treatment and control) by randomizing them in order to "cancel" out any possible uncontrollable outside effects ("lurking variables"). I'm not sure how much of this you have study, but the general saying is "correlation does not imply causation". So while George may notice an increase in the number of birds with the addition of a water source (a positive correlation), it does not imply causation.

Answer 20

You can do some kind of proving causation in an observational study, but the criteria to do so are very rigorous.