Lesson 5 Lies and Statistics Practice Understanding

Learning Focus

Clarify differences between residuals and correlation coefficients.

Use precise statistical language to discuss uses of data.

What do correlation coefficients, linear regressions, and residuals really tell us about bivariate data?

Open Up the Math: Launch, Explore, Discuss

Decide whether each statement is:

Sometimes true
Always true
Never true

Explain your answer with reasoning and examples.

1.

The slope of the linear regression line can be calculated using any two points in the data.

2.

If the correlation coefficient for a set of data is $0$ , then the line of best fit is horizontal.

3.

The sum of the residuals for the line of best fit is $0$ .

4.

If the correlation coefficient is very large, then there must be an outlier in the data.

5.

A negative correlation coefficient means that the data points are very random and don’t really fit a linear model.

6.

A negative residual means that the regression line is very far from the actual data point.

7.

If the correlation coefficient is positive, then the slope of the line of best fit will probably be positive.

8.

If there is a perfect correlation between variables in the data, then the correlation coefficient is $1$ .

9.

To find the value of a residual for a point, $(a, b)$ , given a line of best fit, $f (x)$ :

Find $f (a)$ .
Find $b - f (a)$ .
If the answer is positive, then the point is above the line.
If the answer is negative, then the point is below the line.

10.

The larger the residual for a given point, the further away the point is from the line of best fit.

11.

If there is a perfect correlation between two variables $a$ and $b$ , then either $a$ caused $b$ or $b$ caused $a$ .

Ready for More?

Choose a statistical question about a relationship between two numeric variables that you are interested in, and collect data to answer the question. Your data should include at least $10$ points. Some topics you may find interesting are sports, health and fitness, and hobbies. There is data to be collected about nearly everything.

Once you have some data:

a.

Plot your data to see if there appears to be a relationship.

b.

Describe the strength and direction of the relationship using the correlation coefficient.

c.

If appropriate, find a linear regression for the data, and describe what the regression equation means in context.

Takeaways

Things to remember about correlation coefficients, residuals, and regression lines:

Lesson Summary

In this lesson, we clarified the meaning of correlation coefficients and residuals and how they relate to regression lines.

Retrieval

Determine if the function is linear, exponential, or neither.

1.

$f (n) = f (n - 1) \cdot 9, f (1) = - 7$

A.

linear

B.

exponential

C.

neither

2.

$x$	$g (x)$
$2$	$5$
$3$	$7$
$4$	$10$
$5$	$14$

A.

linear

B.

exponential

C.

neither

3.

$h (x) = 5 (x - 1) - 73$

A.

linear

B.

exponential

C.

neither