Lecture26

Lecture26 - STAT 350 Lecture 26 11.2 Inferences about the...

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Unformatted text preview: STAT 350 Lecture 26 11.2 Inferences about the slope Final Exam Date: May 3, 2011 (Tuesday) Time: 7 PM 9 PM Where: EE 270 Crib sheet: three pages, 1-sided, handwriSen Simple Linear Regression Model The true regression line as a model: yi = + xi + ei In this model: ei is assumed to follow a normal distribuUon with mean 0 and standard devia,on all ei 's are assumed independent of each other Our Goal: EsUmaUng , , and EsUmaUng the slope and intercept A point esUmate of is b A point esUmate of is a EsUmaUng The model assumes ei ~ N(0, ) Recall: Why n 2? We calculaUng two esUmates, a and b, hence we lose 2 df Example Revisited (ex 20 on page 126) Example Revisited (Lecture 21 - ex 20 on page 126) 1) Find point esUmates of the slope and intercept of the populaUon regression line. 2) What is the equaUon of the esUmated true regression line? 3) Find a point esUmate of the error standard deviaUon . 4) What proporUon of the observed variaUon in y can be aSributed to the simple linear regression relaUonship between x and y? Example Revisited (Lecture 21 - ex 20 on page 126) 1) Find point esUmates of the slope and intercept of the populaUon regression line. 2) What is the equaUon of the esUmated true regression line? y = -15.245 + 0.094x + e Example Revisited (Lecture 21 - ex 20 on page 126) 3) Find a point esUmate of the error standard deviaUon . Answer: note: Root MSE entry on the SAS output 4) What proporUon of the observed variaUon in y can be aSributed to the simple linear regression relaUonship between x and y? Answer: R-square = 0.4514 11.2 Inferences about the slope NotaUon: : the slope of the populaUon regression line b: the slope of the sample regression line b is a point es,mate of , just like is a point esUmate of p is a point esUmate of Goal: Sta,s,cal Inference A Confidence Intervals for TesUng Hypotheses about Sampling distribuUon of the slope b b follows a normal distribuUon Standardize b: Sampling distribuUon of the slope, b Standard Z distribuUon when is known: When is unknow, replacing b by sb gives us a familiar t distribuUon with df = n 2 Confidence interval for The confidence interval for is: b (t crit)sb Procedures: Calculate esUmates b and sb Determine confidence level Find t crit from the table Interpret the interval for the true slope Example 11.4 (same data as 11.2) Calculate 95% CI for : b (t crit)sb Step 1: Calculate b Example 11.4 (same data as 11.2) Calculate 95% CI for : b (t crit)sb Step 2: Calculate sb Example 11.4 (same data as 11.2) Calculate 95% CI for : b (t crit)sb Step 3: find t* and the CI 95% CI: -0.92 (2.16)*(0.146) -0.920.315 (-1.233, -0.603) Hypotheses tesUng for H0: = 0 Ha: 0 (can do < or >) Test staUsUcs is: Based on t distribuUon with df = n 2 Usually want to test H0: = 0 Why? Model UUlity Test H0: =0 (no useful linear relaAonship) Ha: 0(can do < or >) Test staUsUcs is: Based on t distribuUon with df = n 2 Learn to read b and sb from SAS output Example Revisited (ex 20 on page 126) Example Revisited (ex 20 on page 126) Perform a model uUlity test H0: =0 (the model is not useful) Ha: 0 (there is a useful linear relaUonship between the variables) Example Revisited (ex 20 on page 126) Perform a model uUlity test H0: =0 (the model is not useful) Ha: 0 (there is a useful linear relaUonship between the variables) SAS output b=0.09424 and sb=0.02215 Example Revisited (ex 20 on page 126) Perform a model uUlity test H0: =0 (the model is not useful) Ha: 0 (there is a useful linear relaUonship between the variables) SAS output b=0.09424 and sb=0.02215 Test staUsUc t=b/sb = 0.009424/0.002215=4.25 df=22, and pvalue=0; so reject H0 in favor of Ha. Conclusion: there is a useful linear relaUonship between the variables. Example--Height and Weight The following data set gives the average heights and weights for American women aged 30-39 (source: The World Almanac and Book of Facts, 1975). Total observaUons 15. SAS Code proc reg data=example; model weight = height; plot weight*height; run; Note the `clb' opUon will produce a confidence interval for b in addiUon to the hypothesis test Example--Height and Weight SAS output--Height and Weight Using the SAS output for inference Construct a 95% confidence interval for Test whether or not there is a significant linear relaUonship (H0: = 0) SAS Code proc reg data=example; model weight = height / alpha=0.05 clb; plot weight*height; run; Note the `clb' opUon will produce a confidence interval for b in addiUon to the hypothesis test SAS output Equivalent CorrelaUon test H0: = 0 H0: = 0 (the model is not useful) Ha: 0 Ha: 0 Test staUsUcs is: Where r is the sample correlaUon coefficient SUll based on t distribuUon with df = n 2 Example (ex 15 on page 508) The value of the sample correlaUon coefficient is 0.449 for the n=14 observaUons on x = hydorgen content and y = gas porosity. Carry out a test at significance level 0.05 to decide whether these two variables are linearly related in the populaUon from which the data was selected. Example (ex 15 on page 508) SoluUon (=0.05): H0: = 0 (the model is not useful) Ha: 0 Also, from the data that r=0.449 and n=14 df=12, p-value=0.11 > 0.05 Conclusion: no evidence of linear relaUonship. => do not reject H0 SAS output Weight and Height Example (Revisit) Carry out a significant test to decide if weight and height are linearly related: H0: = 0 (no linear relaAonship) Ha: 0 =0.05 (default) Also, from the SAS output that R2=0.9910 and n=15 df=13, p-value<0.001 => reject H0 Conclusion: strong evidence of linear relaUonship. ...
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