# Regression inference confidence intervals

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Focus FoxWhat is a regression line?

What is the equation of a regression line in variables?

What is a residual?

What is a residual plot?

What is a normal probability plot?

Inference in RegressionPopulation regression line uses all the observations.

µ = α + βx

Sample regression line (estimated regression line) uses only the observations in a SRS.

y = a + bx

Sampling distribution of b – slope Shape: distribution roughly symmetric, Normal Probability Plot (z-score) is linear – close to NormalCenter: mean of sampling distribution is close to slope of population regression lineSpread: standard deviation of sampling distribution is close to actual standard deviation in population.

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Inference in RegressionConditions for Regression Inference - 5Linear: examine scatterplot to check overall pattern is roughly linear, residuals should center on 0, look for curved pattern in residual plot

Independent: look for random assignment or experiment, if sampling without replacement – check 10% condition

Normal: make a stemplot, histogram, or Normal probability plot of residuals and check for skewness or departure from Normality

Equal Variance: the scatter in the residuals (above and below the line) should be roughly the same

Random: were the data produced by random sampling or random assignment

Inference in RegressionMrs. Barrett’s class did an experiment dropping helicopters from various heights. 14 helicopters at each of 5 drop heights in centimeters. Teams of students released the 70 helicopters in a predetermined random order and measured the flight times in seconds. The class used Minitab to carry out a least-squares regression analysis for these data. A scatterplot, residual plot, histogram, and Normal probability plot of the residuals are on the next slide.

Check whether the conditions for performing inference about the regression model are met.

Inference in Regression

Inference in RegressionStandard deviation of a sample describes the size of the typical prediction error and is found using:

s = =

This standard deviation is found for you in most computer outputs:

Estimate β, what is the meaning of α, typical prediction error??

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Inference in RegressionIt is possible to do inference on any of the three parameters in the regression model, , β, or α, but we will most commonly inquire about inference of slope or the β parameter.

When estimating a spread from a sampling distribution, we use the standard error of the slope:

SEb =

Since we are using the average of the sample slopes, we will use a t* score, meaning table B. To standardize the scores:

t =

With degrees of freedom = n – 2

Inference in RegressionThese components allow us to find a confidence interval for the prediction of slope β from our sample data. statistic ± critical value • standard deviation of statistic

b ± t*SEb

New type of inference: t-interval for the slope of a Least-Squares Regression LineConfidence interval: b ± t*SEb

Standard error of slope: SEb = (how much slope in sample regression line typically varies from slope in population regression line)

t* is critical value from table BDegrees of Freedom = n – 2

Inference in RegressionHelicopter Activity Computer Output:

a. Identify the standard error of slope SEb from the computer output, interpret this value in context.

Inference in RegressionHelicopter Activity Computer Output:

b. Find the critical value for a 95% confidence interval for the slope of the true regression line. Then calculate the confidence interval.

Inference in RegressionHelicopter Activity Computer Output:

c. Interpret the confidence in context

d. Interpret the meaning of “ 95% confident” in context.

Inference in RegressionPg. 749-750