Use the reference below when answering the questions

We calculate a value called the Pearson Correlation Coefficient r.

We then compare that value of r to a table value.

If the absolute value of our calculated Pearson r, denoted , is greater than the table value, linear correlation exists between the variables.

If linear correlation can be determined by this test, then we can find the equation for the regression line. The line that on average is closest to all the plotted points.

Linear correlation between two variables means that they share a mathematically predictable linear relationship.Linear correlation can be positive or negative.

It is positive when as one variable increases, the other also increases.

It is positive when as one variable decreases, the other also decreases.

It is negative when as one variable increase, the other decreases.

Positive linear correlation examples:

As the temperature increases, ice cream sales increase.

As the percentage of salt in water increases, buoyancy increases.

As the temperature decreases, air conditioning costs decrease.

Negative linear correlation examples:

As time running increases, a person's energy decreases.

As the speed of a car decreases, time to the destination increases.

As absences increase, grades decrease.

The calculated Pearson r will be a positive or negative number, indicating the possibility of positive or negative linear correlation.

Below is an example of calculating the Pearson r, confirming linear correlation, and finding the equation for the regression line.

The calculations are not difficult, but they are quite tedious and time consuming.

Later in the lesson, you will see that our focus is not going to be on you grinding through these long calculations, but rather on you being able to interpret the results.

Consider the ordered pairs (3, 4), (7, 8), (5, 6) and (1, 2).

We are going to test for linear correlation at a 95% confidence level.

We will calculate the Pearson |r|. Iftable value, then we have linear correlation and will find the regression line.

Note that in math, the symbol instructs you to add stuff.

To calculate the Pearson r, we need 5 values:

x the sum of the x-components = 3 + 7 + 5 + 1 = 16

y the sum of the y-components = 4 + 8 + 6 + 2 = 20

(xy) the sum of the multiples of the x and y-components =

(3x4) + (7x8) + (5x6) + (1x2) = 100

x² the sum of the squares of the x-components =

3^2 +7^2 +5^2 +1^2 = 84

y² the sum of the squares of the y-components =

4^2+ 8^2 +6^2 +2^2 = 120

Label the following four statements as examples of positive or negative correlation:

1) As snowfall totals increase, the number of people driving decreases.

2) As the wattage of light bulbs increase, the light output increases.

3) The more you pay off a loan, the less debt you will have.

4) The more time you spend on a treadmill, the more calories you will burn.

5) Does a Pearson r of -0.246 indicate the possibility of positive or negative correlation?

6) Imagine that 10 ordered pairs have a calculated Pearson r of 0.572.

With a confidence level of 95%, is linear correlation confirmed or rejected?

7) Assume that linear correlation has been confirmed by the Pearson r test.

The slope (m) has been calculated to equal 1/3.

The y-intercept (b) has been calculated to be 4.

What is the equation of the regression line?

8) From 2000 to 2009 there was a strong positive correlation between the amount of margarine consumed and the divorce rate in the state of Maine.

Both margarine consumption and the divorce rate declined in correlated manner.

a) Is it reasonable to conclude that the people of Maine eating less margarine caused fewer Maine couples to divorce?

b) Is it reasonable to conclude that divorces cause people in Maine to consume less margarine?