Chapter 12 Inference for Linear Regression

Reminder of Linear Regression First thing you should do is examine your data… First thing you should do is examine your data… Look at your scatterplot. Does it appear linear? Are there outliers? What direction is it going in? Is there a strong relationship? Look at your scatterplot. Does it appear linear? Are there outliers? What direction is it going in? Is there a strong relationship? LSR: y-hat = a + bx LSR: y-hat = a + bx a = y-intercept; b= slope a = y-intercept; b= slope Slope is the rate of change in y for every one x. Slope is the rate of change in y for every one x.

Statistics versus parameters a and b are statistics (estimates of y-intercept and slope). a and b are statistics (estimates of y-intercept and slope). α and β are unknown parameters α and β are unknown parameters a is an unbiased estimator of α and b is an unbiased estimator of β a is an unbiased estimator of α and b is an unbiased estimator of β

We are interested in β We are going to look at inference for β (slope). We are going to look at inference for β (slope). Confidence Intervals and Hypothesis tests. Confidence Intervals and Hypothesis tests.

Confidence Intervals These will be t-tests These will be t-tests What is the basic formula for confidence intervals? What is the basic formula for confidence intervals? Estimate +/- margin of error Estimate +/- margin of error Estimate +/- t-statistic*Standard Error Estimate +/- t-statistic*Standard Error For inference for the true mean slope (β) For inference for the true mean slope (β) b +/- t*(SE) b +/- t*(SE)

Standard Error and DF You will either be given this information or you can get your calculator to give it to you! You will either be given this information or you can get your calculator to give it to you! Degrees of freedom = n – 2 Degrees of freedom = n – 2 Why? Why?

Computer Output Look with me pg 770 Look with me pg 770 Remember, under coefficient… Remember, under coefficient… Constant = y-intercept Constant = y-intercept Variable definition = slope Variable definition = slope Standard error is the second row under STDev Standard error is the second row under STDev

Hypothesis Tests Generally, Generally, H 0 = 0 H 0 = 0 This says that the true slope is zero, which means there is no change in y. This can be different if the context of the problem would mean that no change is not zero… This says that the true slope is zero, which means there is no change in y. This can be different if the context of the problem would mean that no change is not zero…

Calculator! Put your data in List 1 and List 2 Put your data in List 1 and List 2 In your calculator, you go to LinRegTTest under Stat, Test In your calculator, you go to LinRegTTest under Stat, Test

Example How well does the number of beers a student drinks predict his or her blood alcohol level? Sixteen student volunteers at Ohio State University drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC). How well does the number of beers a student drinks predict his or her blood alcohol level? Sixteen student volunteers at Ohio State University drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC).

The Data Student Beers BAC Student Beers BAC

Conditions Observations are independent Observations are independent You don’t observe the same person multiple times You don’t observe the same person multiple times The true relationship is linear The true relationship is linear Check residual plot for scatter. Look at scatter plot. Check residual plot for scatter. Look at scatter plot.

Conditions Continued The spread is uniform The spread is uniform The residual plot does not have a cone like appearance. The residual plot does not have a cone like appearance. The residuals have a normal distribution. The residuals have a normal distribution. Graph residuals Graph residuals

Residuals Since almost all the conditions deal with residuals, we should probably review Since almost all the conditions deal with residuals, we should probably review Residual = observed – predicted Residual = observed – predicted y – (y-hat) y – (y-hat) In you calculator: Define L3 as L2 – Y1(L1) In you calculator: Define L3 as L2 – Y1(L1) You can look at a scatter plot of L1 vs. L3 to see residual plot. You can look at a scatter plot of L1 vs. L3 to see residual plot. To determine normality, look at a histogram of L3 To determine normality, look at a histogram of L3

Example of Ohio State University Check your conditions for the previous problem. Check your conditions for the previous problem. Let’s finish the problem now. Let’s finish the problem now.

Example of when H 0 is not β = 0

Homework: Homework: Read Chapter 12. Do questions #9, 10, 14, 18, MC (explain) Read Chapter 12. Do questions #9, 10, 14, 18, MC (explain)

You have now finished all of your AP Statistics Course work!!!!!!!