regression inference tests

1. Inference in Regression
We can also complete a significance test to determine if a specified value of
β is plausible.
Null Hypothesis has the form H0: β = hypothesized value
To run the test, standardize b to find test statistic
test statistic = statistic – parameter
standard deviation of statistic
To find P-value use t distribution with degrees of freedom = n – 2
New type of Inference:
t test for the slope of the Population Regression Line
t = b – β0 Ha: β > hyp value Ha: β < hyp value Ha: β ≠ hyp value
SEb

2. Inference in Regression
H0: β = 0, a regression line with a slop of 0 is horizontal
H0 says that there is no linear relationship between x and y in the population;
the linear regression of y on x is of no value for predicting y
Regression output from statistical software usually gives t and its two-sided
P-value for a test of H0: β = 0. To find the one-sided P-value divide the
given P-value by 2.

3. Inference in Regression
Infants who cry easily may be more easily stimulated than others. This may
be a sign of higher IQ. Child development researchers explored the
relationship between the crying of infants 4 to 10 days old and their later IQ
test scores. A snap of a rubber band on the sole of the foot caused the infants
to cry. The researchers recorded the crying and measured its intensity by the
number of peaks in the most active 20 seconds of crying. They later
measures the children’s IQ at age three years using the Stanford-Binet IQ
test.

4. Inference in Regression
a. Using the scatterplot of the data with the least-squares line added,
describe what this graph tells you about the relationship between these two
variables.

5. Inference in Regression
What is the equation of the least-squares regression line for predicting IQ at
age 3 from the number of crying peaks?
Interpret the slope and y-intercept of the regression line in context.

6. Inference in Regression
Do theses data provide convincing evidence that there is a positive linear
relationship between crying counts and IQ in the population of infants?
Carry out an appropriate test.

7. Inference in Regression
Notice the T value in the table. It is our test statistic.
Notice the P-Value in the table. It is double the P-value in our test, because
our test is one-sided.
Remember that confidence intervals give you a little more information than
tests do. They tell you not just that is greater, but where and how much
greater.

8. Inference in Regression
Our bodies have a natural electrical field that is known to help wounds heal.
How does changing the field strength affect healing? A series of experiments
with frogs investigated this question. In one experiment, the two hind limbs
of 12 frogs were assigned at random to either experimental or control groups.
The electrical field in the experimental limbs was reduced to zero by
applying a voltage. The control limbs were left alone. Researchers
measured the rates at which new cells closed a razor cut in each limb in
micrometers per hour.
An examination of the data suggests that the conditions for performing
regression inference are met. A 95% confidence interval for the slope β of
the true regression line for predicting healing rates in frogs’ limbs when
exposed to zero voltage from healing rates in their limbs with normal
electrical fields is (-0.42, 0.72). Based on the interval, what conclusion
should we draw from a test of H0: β = 0 versus Ha; β ≠ 0 at the α = 0.05
significance level?

9. Inference in Regression
Using graphing calculators to run a t test for regression
Enter NEA change in L1 and y values in L2
STAT – TESTS – LinRegTTest
TI-84 can also do a LinRegTInt
NEA change -94 -57 -29 135 143 151 245 355 392 473 486 535 571 580 620 690
Fat gain 4.2 3.0 3.7 2.7 3.2 3.6 2.4 1.3 3.8 1.7 1.6 2.2 1.0 0.4 2.3 1.1