Welcome to class today!Chapter 12 summary sheet

Jimmy Fallon videohttps://www.youtube.com/watch?

v=bG1Q2SkpttI

CORRECT HOMEWORKChapter 3 Review

12.1: INFERENCE FOR LINEAR REGRESSION

Notes

Population vs. Sample

• Population regression line (True regression line) – uses entire population

• Sample regression line – uses only sample data

Inference for linear regression (How does the slope of the sample regression line relate to the slope of the population regression line.

Sampling Distribution of b (slope)

Confidence intervals and significance tests about the slope of the population regression line are based on the sampling distribution of b, the slope of the sample regression line.

Conditions

1. Linear2. Independent3. Normal4. Equal Variance5. Random

Linear

Examine the scatterplot to check that the overall pattern is roughly linear. Look for curved patterns in the residual plot. Check to see that the residuals center on the “residual = 0” line at each x-value in the residual plot.

Independent

Look at how the data were produced. Random sampling and random assignment help ensure the independence of individual observations. If sampling is done without replacement, remember to check that the population is at least 10 times as large as the sample (10% condition).

Normal

Make a stemplot, histogram, or Normal probability plot of the residuals and check for clear skewness or other major departures from Normality.

Equal Variance

Look at the scatter of the residuals above and below the “residual = 0” line in the residual plot. The amount of scatter should be roughly the same from the smallest to the largest x-value.

Random

See if the data were produced by random sampling or a randomized experiment.

ExampleMany people believe that students learn better if they sit closer to the front of the classroom. Is this true? A teacher randomly assigns students to seat locations in his classroom for a particular chapter and recorded the test score for each student at the end of the chapter? Here are the results:Row #1: 76, 77, 94, 99Row #2: 83, 85, 74, 79Row #3: 90, 88, 68, 78Row #4: 94, 72, 101, 70, 79Row #5: 76, 65, 90, 67, 96Row #6: 88, 79, 90, 83Row #7: 79, 76, 77, 63

Graphs of the data

Continue Examplea. State the equation of the least-squares regression line. Define any variable you use.b. Interpret the slope, y-intercept, and standard deviation of the residuals.

Standard Error of the Slope

We don’t know σ for the population regression line. So we estimate it with the standard deviation of the residuals, s. Then we estimate the spread of the sampling distribution of b with the standard error of the slope

T distribution

What happens if we transform the values of b by standardizing? Since the sampling distribution of b is Normal, the statistic

t Interval for the Slope of a Least-Squares Regression Line

When the conditions for regression inference are met, a level C confidence interval for the slope βof the population (true) regression line isb ± t* SEb

In this formula, the standard error of the slope is

and t* is the critical value for the t distribution with df = n - 2 having area C between -t* and t*.

Example

a. Identify the standard error of the slope SEb from the computer output. Interpret this value in context.b. Calculate the 95% confidence interval for the true slope. c. Interpret the interval from part (b)d. Based on your interval, is there convincing evidence that seat location affects scores?

Your turn!For their second-semester project, two AP statistics students decided to investigate the effect of sugar on the life of cut flowers. There went to the local grocery store and randomly selected 12 carnations. All the carnations seemed equally healthy when they were selected. When the students got home, they prepared 12 identical vases with exactly the same amount of water in each vase. They put one tablespoon of sugar in 3 vases, and three tablespoons of sugar in 3 vases, they put no sugar in the remaining 3 vases. After the vases were prepared and placed in the same location, the students randomly assigned one flower to each vase and observed how many hours each flower continued to look fresh. Here is the data:

The Data and Question

a. Construct and interpret a 99% confidence interval for the slope of the true regression line.b. Would you feel confident predicting the hours of freshness if 10 tablespoons of sugar are used? Explain.

The Graphs

PG. 759 #2, 6, 8, 10Homework Due Monday

Exit Ticket

Create a creative saying to help remember the conditions that are required for performing inference tests for linear regression.