Inference in RegressionWe 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 statistictest statistic = statistic – parameter

standard deviation of statisticTo 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

Inference in RegressionH0: 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.

Inference in RegressionInfants 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.

Inference in Regressiona. 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.

Inference in RegressionWhat 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.

Inference in RegressionDo 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.

Inference in RegressionNotice 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.

Inference in RegressionOur 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?

Inference in RegressionUsing 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