1. 10-3 Inferences

2. 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
The Idealized model….
Certain assumptions must be met:
(x,y) is a random sample from the population
For each fixed x, the y has a normal distributions.
All the y distributions have the same variance
Other books use β0 and β1 for a and b

3. Testing ρ 2. Compute the test statistic
i.e. Testing whether there is or is not a linear correlation
1. Set your hypotheses
HO: ρ = HA: ρ > 0, ρ < 0, ρ ≠ 0
2. Compute the test statistic

4. Testing ρ 2. Compute the test statistic
i.e. Testing whether there is or is not a linear correlation
1. Set your hypotheses
HO: ρ = HA: ρ > 0, ρ < 0, ρ ≠ 0
2. Compute the test statistic
d.f. = n – 2
Logic dictates how many data sets (n)?

5. Testing ρ 2. Compute the test statistic 3. Find the P-value
i.e. Testing whether there is or is not a linear correlation
1. Set your hypotheses
HO: ρ = HA: ρ > 0, ρ < 0, ρ ≠ 0
2. Compute the test statistic
3. Find the P-value
4. Compare to α and conclude.
5. State your conclusion.
What is the purpose? Seeing if there is a linear correlation of any type, NOT determining causation…

6. Measuring Spread
What if you feel there is a linear correlation, but are not sure how “good” it is. That is, while the line maybe touches two points, some other points are substantially off….
The line is a model. The model assumes that the means of the distributions fall on the line even though individual data points may not…

7. Measuring Spread Error can be determined a number of ways
Method 1: Using residual

8. Measuring Spread Error can be determined a number of ways
Method 1: Using residual
Where ŷ = ax + b, and n > 3
Use this one!!

9. Measuring Spread (cont)
Method 2: Find 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.

10. Measuring Spread (cont)
Method 2: This will look familiar…. Based on
n ≥ 3 data pairs, after finding ŷ use

11. 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

12. Measuring Spread (cont)
Method 3: Using a student t distribution, you will test the population slope (β)
Based on a null hypotheses of β = 0 and an alternate like before (greater, less than or not equal to..)
What will the test statistic simplify to be?

13. Measuring Spread (cont)
Method 3: Using a student t distribution, you will test the population slope (β)
Based on a null hypotheses of β = 0 and an alternate like before (greater, less than or not equal to..)
What will the test statistic simplify to be?
d.f. = n-2

14. Measuring Spread (cont)
Method 3: Therefore, the confidence interval will be

15. Calculator
Your calculator can find standard error, and can also do a linear regression test (callced a LinRegTTest)

16. Problem
Let x be a random variable that represents the batting average of a professional baseball player. Let y be a random variable that represents the percentage of strikeouts of a professional baseball player. A random sample of 6 professional baseball players gave the following information: x
0.328
0.290
0.340
0.248
0.367
0.269 y
3.2
7.6
4.0
8.6
3.1
11.1

17. Given the following
a) use a 5% confidence level to test the claim that ρ≠0.
b) Find Se, a and b
c) Find the predicted percentage of strikeouts for a player with an x = batting average
d) Find an 80% confidence interval for y when x = 0.300
e) Use a 5% level of significance ot test the claim that β≠0
f) Find a 90% confidence interval for β and interpret its meaning.