Confidence interval question

Question

A 90 percent confidence interval for the slope of a regression line is determined to be (-0.181, 1.529). Which of the following statements must be true?

a) The correlation coefficient of the data is positive

B) sum of residuals based on regression line is positive

c) A scatterplot of data shows linear pattern

d) Slope of the sample regression line is 1.348

e) the slope of sample regression line is 0

The answer is a. I am stumped and don't see how that is the answer at all.

Answer 1

I'm just adding to what was already said in case anything I add helps clarify further (I just added explanations to each answer choice explaining why it was correct/incorrect.):

A 90 percent confidence interval for the slope of a regression line is determined to be (-0.181, 1.529). Which of the following statements must be true?

a) The correlation coefficient of the data is positive - This is the correct answer because it specifically says the correlation coefficient of the DATA is positive (meaning only from the sample not from the population we are testing a claim about. We know the slope from the sample would have been 0.674 because that is the number directly in the middle of the confidence interval. Since the slope from the sample is positive, that means the correlation coefficient calculated from the sample must also be positive.

B) sum of residuals based on regression line is positive- The sum of the residuals is 0.

c) A scatterplot of data shows linear pattern - We do not have enough information to determine this.

d) Slope of the sample regression line is 1.348 As stated in the explanation for A, the slope from the sample regression line is .674 - which was calculated by adding the endpoints of the interval and then dividing by 2... the error that occurred here was that this is the sum of the endpoints of the interval without dividing by 2.

e) the slope of sample regression line is 0 - While 0 is contained in the interval (meaning it is a possible value for the true slope of the population regression line), we have already said that the slope of the sample regression line was .674

If you have any follow-up questions, please let me know! Taught AP Stats for many years and I love this stuff!

Answer 2

So first, let's eliminate the other options. Obviously, D and E aren't correct because 1.348 isn't the midpoint of the confidence bounds and the sample regression line isn't 0: it's just tested to be zero. Because of this test, we don't need to see a linear pattern in the data, so C is out as well.

By definition, the sum of the residuals is 0. Note, B does not say "squared residuals;" it just says residuals. So that can't be positive.

That leaves us with A. Now there is a positive slope that's seen in this data even though that slope isn't significant. If the slope is positive, the correlation coefficient is also positive. Remember that β = r•s_x÷s_y. Since the standard deviations for x and y are both positive, then since the β>0, r>0. This makes choice A as the correct answer.

Answer 3

b) and e) are false

sum of residuals always = 0, eliminating b)

slope is not necessarily 0, although possibly zero, eliminating e)

that leaves a), c) and d)

the point estimate for the slope is not 1.348, but (1.529- .181)/2 = 1.348/2 = .674

so eliminate d)

that leaves a) or c)

the relationship may approximate linear but not necessarily be linear, that eliminates c)

You can run a regression analysis and get a correlation coefficient even if the sample is nonlinear.

that leaves only a), by process of elimination

in a time pressure test, stop there and go with a)

or look closer at a)

the point estimate is positive

and the slope is mostly positive in the CI, Confidence Interval,

with, not fully 90% confidence but from (0, 1.348) it's positive which is close to 90%, maybe 82.23% or so

use a z calculator or z tables

the point estimate, +.674, for slope is definitely positive

BUT, a) doesn't say the slope is necessarily positive, it just says the correlation coefficient is positive, r^2 >0, and r^2 is always positive, as any number squared is positive, except zero. odds of exactly zero are infinitesmal or close to zero. Still, the correlation coefficient =r, not r^2. r^2 = the coefficient of determination, sometimes confused with the correlation coefficient = r. the test maker may even have confused them.

the linear regression line is y=ax +b

where the sign of a is the same as the sign of the correlation coefficient = covxy/varxvary = the point estimate for the slope which is necessarily positive, = +.674, in the given problem

"none of the above" is not an option. Usual test instructions for multiple choice questions say choose the "best" answer, which sometimes is false, just less false than all the alternatives. a) is clearly more true than any of the other 4 alternatives

but I defer to any statistics professors out there, or statistics majors, or someone who took a recent course in regression analysis