regression inference confidence intervals
DESCRIPTION
Conditions & Regression Inference using Confidence IntervalsTRANSCRIPT
![Page 1: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/1.jpg)
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?
![Page 2: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/2.jpg)
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.
^
![Page 3: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/3.jpg)
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
![Page 4: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/4.jpg)
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.
![Page 5: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/5.jpg)
Inference in Regression
![Page 6: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/6.jpg)
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??
^
![Page 7: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/7.jpg)
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
![Page 8: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/8.jpg)
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
![Page 9: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/9.jpg)
Inference in RegressionHelicopter Activity Computer Output:
a. Identify the standard error of slope SEb from the computer output, interpret this value in context.
![Page 10: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/10.jpg)
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.
![Page 11: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/11.jpg)
Inference in RegressionHelicopter Activity Computer Output:
c. Interpret the confidence in context
d. Interpret the meaning of “ 95% confident” in context.
![Page 12: Regression inference confidence intervals](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f2d99b4af9f6c4f8b47c8/html5/thumbnails/12.jpg)
Inference in RegressionPg. 749-750