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### Course: Statistics and probability > Unit 5

Lesson 4: Least-squares regression equations- Introduction to residuals and least squares regression
- Introduction to residuals
- Calculating residual example
- Calculating and interpreting residuals
- Calculating the equation of a regression line
- Calculating the equation of the least-squares line
- Interpreting slope of regression line
- Interpreting y-intercept in regression model
- Interpreting a trend line
- Interpreting slope and y-intercept for linear models

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# Interpreting slope of regression line

We look at an example scatter plot showing the relationship between hours studying and test scores. We interpret scatter plots, investigate the relationship between variables, interpret the slope of a trend line, and make predictions based on the trend line's slope and intercept.

## Want to join the conversation?

- how do you find the slope if they do not give it to you?(6 votes)
- The slope of a least squares regression can be calculated by m = r(SDy/SDx).

https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/calculating-the-equation-of-a-regression-line

In this case (where the line is given) you can find the slope by dividing delta y by delta x. So a score difference of 15 (dy) would be divided by a study time of 1 hour (dx), which gives a slope of 15/1 = 15.(6 votes)

- How do you interpret the slope when it's a fraction??(3 votes)
- So, first things first we need to know what slope is. Slope is the change in y/change in x; the same thing as rise/run.

Here is an example:

Lets say you have a equation that says y=1/4x+2. Its pretty simple from there. So, we know in the slope intercept formula (y=mx+b) we know that m=slope and b=y intercept. So for the equation I gave you m=1/4 and b=2. So, from the y-intercept (which is 2) you move 4 spaces to the right and 1 space up.

Hope that helps!:)(5 votes)

- What does slope mean?(4 votes)
- rise (y_final - y_initial) over run (x_final - x_initial)(2 votes)

- So regression line is essentially just "line of best fit"(1 vote)
- Yes, with the idea coming from OLS (Ordinary Least Squares). Its goal is to reduce the residuals of errors to produce the best-fit line.(2 votes)

## Video transcript

- [Instructor] Liz's math test
included a survey question asking, how many hours
students that spent studying for the test. The scatter plot and trend line below show the relationship between how many hours
students spent studying and their score on the test. The line fitted to the model, the line fitted to model the data has a slope of 15. So, the line they're talking about is right here. So this is the scatter
plot, this shows that some student who spent
some time in between half an hour and an hour studying got a little bit less
than a 45 on the test. This student here, who got a
little bit higher than a 60 spent a little under two hours studying. This student over here,
who looks like they got like a 94, or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices what's the best
interpretation of this slope? Well, if you think this line
is indicative of the trend, and it does look like that
from this scatter plot, that implies that, roughly every extra hour that you study is going to improve your score by 15. You could say, on average,
according to this regression. So, if we start over here and we were to increase by one hour our score should improve by 15. And it does, indeed, look like that. We're going from, we're going in the horizontal direction,
we're going one hour. And then in the vertical
direction we're going from 45 to 60. So that's how I would interpret it. Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement. Or at least that's what we're seeing that's what we're seeing from
the regression of the data. So let's look at which of these choices actually describe something like that. The model predicts that
the student who scored zero studied for an average of 15 hours. No, it definitely doesn't say that. The model predicts that
students who didn't study at all will have an average score of 15 points. No we, we didn't see that. Students, if you take this,
if you believe this model someone who doesn't study at all would get close to would get between 35 and 40 points. So like a 37, or a 38. So, don't like that choice. The model predicts the score will increase 15 points for each additional
hour of study time. Yes, that is exactly what
we were thinking about when we were looking at the model. That's what a slope of 15 tells you. You increase studying time by an hour it increases the score by 15 points. The model predicts that the study time will increase 15 hours for each additional point scored. Well, no. And first of all, the hours is the thing that we use the independent variable and the points being
the dependent variable, and this is phrasing it the other way. And you definitely wouldn't
expect to do an extra 15 hours for each point.