AP/H Statistics Guided NotesMrs. LeBlanc – Perrone

9.3 Inferences for Correlation and RegressionPart 1: Testing ρand the Standard Error of Estimate

Inferences for Correlation and Regression

In Sections 9.1 and 9.2, we learned how to compute the sample correlation coefficient r and the least-

squares line y=a+bx using data from a sample.

o r is only a _____________________________________________________________________

o y=a+bx is only a _____________________________________________________________

o What if we used all possible data pairs?

In theory, if we had the population of all (x, y) pairs, then we could compute the

_________________________________________________________ (Greek letter rho)

and we could compute the __________________________________________________

________________________________________________________________________

Note the following:

Sample Statistic Population Parameter

r ¿¿

a ¿¿

b ¿¿

y=a+bx ¿¿

Requirements for Statistical Inference

o To make inferences regarding correlation and linear regression, we need to be sure that

The set (x, y) of ordered pairs is a random sample from the population of all possible

such (x, y) pairs

For each fixed value of x, the y values have a normal distribution. All of the y

distributions have the same variance, and, for a given x value, the distribution of y values

has a mean that lies on the least-squares line. We also assume that for a fixed y, each x

has its own normal distribution. In most cases the results are still accurate if the

distributions are simply mound-shaped and symmetric and the y variances are

approximately equal.

o ____________________________________________________________________________________

____________________________________________________________________________________

1

Name: ______________________________

Date: ___________

AP/H Statistics Guided NotesMrs. LeBlanc – Perrone

Testing the Correlation Coefficient

The first topic we want to study is the statistical significance of the sample correlation coefficient r.

To do this, we construct a statistical test of , the population correlation coefficient.

How to Test the population correlation coefficient ρ

Let r be the sample correlation coefficient computed using data pair (x , y )

1. Use the null hypothesis ¿¿ (x and y have _______________________________). Use the

context of the application to state the alternate hypothesis (¿¿). State the level of significance α

.

2. Obtain a sample of n≥3 data pairs and compute the sample test statistic

t=¿ with degrees of freedom d . f .=n−2

3. Use the TI-83 or TI-84 to calculate the _____________________

_______________________________________________________________________

4. Conclude the test

If the P-values is ≤α , then reject H 0

If the P-values is ¿α , then fail to reject H 0

5. Interpret the results in the context of your application

2

AP/H Statistics Guided NotesMrs. LeBlanc – Perrone

Example: Testing ρ

Do college graduates have an improved chance at a better income? Is there a trend in the general population

to support the “learn more, earn more” statement? We suspect the population correlation is positive, let’s test

using a 1% level of significance. Consider the following variables: x = percentage of the population 25 or older

with at least four years of college and y = percentage growth in per capita income over the past seven years. A

random sample of six communities in Ohio gave the information shown

Caution: Although we have shown that x and y are positively correlated, we have not shown that an

increase in education causes an increase in earnings.

3

AP/H Statistics Guided NotesMrs. LeBlanc – Perrone

You Try It!

A medical research team is studying the effect of a new drug on red blood cells. Let x be a random variable

representing milligrams of the drug given to a patient. Let y be a random variable representing red blood cells

per cubic milliliter of whole blood. A random sample of n=7 volunteer patients gave the following results.

x 9.2 10.1 9.0 12.5 8.8 9.1 9.5

y 5.0 4.8 4.5 5.7 5.1 4.6 4.2

Use a 1% level of significance to test the claim that ρ≠0.

4

AP/H Statistics Guided NotesMrs. LeBlanc – Perrone

Standard Error of Estimate

Sometimes a scatter diagram clearly ______________________________________________________

between x and y, but it can happen that the points are widely scattered about the least-squares line.

We need a method (besides just looking) for measuring the spread of a set of points about the least-

squares line. There are three common methods of measuring the spread.

o the coefficient of correlation

o the coefficient of determination

o ______________________________________________________

For the standard error of estimate, we use a measure of spread that is in some ways like the

standard deviation of measurements of a single variable. Let _________________________________

________________________________________________ from the least-squares line.

Then y – y is the difference between the y value of the data point (x, y) shown on the scatter diagram

(Figure 9-16) and the yvalue of the point on the least-squares line with the same x value.

The quantity __________ is known as the ___________________. To avoid the difficulty of having

some positive and some negative values, we square the quantity (y – y).

Then we sum the squares and, for technical reasons, divide this sum by n – 2. Finally, we take the

square root to obtain the standard error of estimate, denoted by Se

.

Standard Error of Estimate = ______________________________________________

where y=a+bx and n≥3

Using the TI 83 & TI 84

1. STAT

2. TEST

3. LinRegTTest

The value for Se is given as s5

AP/H Statistics Guided NotesMrs. LeBlanc – Perrone

Example

June and Jim are partners in the chemistry lab. Their assignment is to determine how much copper sulfate

(CuSO4

) will dissolve in water at 10, 20, 30, 40, 50, 60, and 70°C. Their lab results are shown in Table 9-12,

where y is the weight in grams of copper sulfatethat will dissolve in 100 grams of water at x°C. Sketch a scatter

diagram, find the equation of the least-squares line, and compute Se

6

AP/H Statistics Guided NotesMrs. LeBlanc – Perrone

Summary Questions

1. What does testing the population correlation coefficient ρ show?________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2. Complete 9.1 – 9.3 Graphing Calculator Exercises (including the “You Try It”)

“HOT” Question:____________________________________________________________________________________________________________________________________________________________________________________

7