Inferences

10-3

What about making inferences?

r, a and b are the sample test statistics

Sample r becomes population ρ and ŷ = ax + b becomes y = αx + β

Certain assumptions must be met:(x,y) is a random sample from the populationFor each fixed x, the y has a normal distributions. All the y distributions have the same variance CHECK BVD

Testing ρ

i.e. Testing whether there is or is not a linear correlation

1. Set your hypotheses HO: ρ = 0 HA: ρ > 0, ρ < 0, ρ ≠ 0

2. Compute the test statistic

2

r n 2t

1 r

d.f. = n – 2

Logic dictates how many data sets (n)?

Testing ρ

i.e. Testing whether there is or is not a linear correlation

1. Set your hypotheses HO: ρ = 0 HA: ρ > 0, ρ < 0, ρ ≠ 0

2. Compute the test statistic

3. Find the P-value4. Compare to α and conclude.5. State your conclusion.

Measuring Spread

Error can be determined a number of waysMethod 1: Using residual

Where ŷ = ax + b, and n > 3

2 2

e

ˆ(y y) y a y b xyS

n 2 n 2

Use this one!!

Measuring Spread (cont)

Method 2: a confidence interval for y

True y, for a population, has a population slope, a population y intercept, plus some sort of random error.

Therefore, we can create a confidence interval for y that allows us to predict true y.

y x

Measuring Spread (cont)

Method 2: This will look familiar…. Based on n ≥ 3 data pairs, after finding ŷ use

Where ŷ = ax + b, c = confidence level, n = number of data pairs, and Se is the standard error of estimate

ˆ ˆy E y y E

2

c e 22

1 n(x x)E t S 1

n n x x

Least Squares Line

Equation of a line: y = mx + b

In statistics: ŷ = a + bx (Our book) also: ŷ = b0 + b1x

There are formulas for a and b

22

n xy x yb

n x x

a y bx

x

y

sb r

s