Inference for  0 and 1 Confidence intervals and hypothesis tests

Example 1: Relation between leg strength and punting distance? PUNTER LEG DIST punters in American football DIST = average length (feet) of 10 punts LEG = strength of leg (pounds lifted)

Example 2: Relation between state latitude and skin cancer mortality? # State LAT MORT 1 Alabama Arizona Arkansas California Colorado !    49 Wyoming Mortality rate of white males due to malignant skin melanoma from LAT = degrees (north) latitude of center of state MORT = mortality rate due to malignant skin melanoma per 10 million people

Example 3: Relation between use of amphetamines and food consumption? X = Amphetamine dose (mg/kg) rats randomly allocated to dose of amphetamine (saline (0), 2.5, and 5.0 mg/kg) Y = amount of food (grams of food consumed per kilogram of body weight) in following 3-hour period

Point estimates b 0 and b 1 The b 0 and b 1 values vary. They depend on the particular (x i, y i ) sample obtained.

Assumptions about error terms  i E(  i ) = 0  i and  j are uncorrelated Var(  i ) =  2 (New!!)  i are normally distributed NOTE: All results thus far (such as least squares estimates, Gauss-Markov Theorem, and mean square error) only depend on first three assumptions. Today’s results depend on normality of the error terms.

Sampling distribution of b 1 b 1 is normally distributed Providing error terms  i are normally distributed: with mean 1 and variance

Recall: Confidence interval for  using  Sample estimate ± margin of error

Confidence interval for 1 using b 1 Sample estimate ± margin of error

Recall: Hypothesis testing for  using  The null (H 0 :  =  0 ) versus the alternative (H A :  ≠  0 ) Test statistic P-value = How likely is it that we’d get a test statistic t* as extreme as we did if the null hypothesis is true? The P-value is determined by comparing the test statistic t* to a t-distribution with n-1 degrees of freedom.

Hypothesis testing for 1 using b 1 The null (H 0 : 1 = ) versus the alternative (H A : 1 ≠ ) Test statistic P-value = How likely is it that we’d get a test statistic t* as extreme as we did if the null hypothesis is true? The P-value is determined by comparing the test statistic t* to a t-distribution with n-2 degrees of freedom.

Sampling distribution of b 0 b 0 is normally distributed Providing error terms  i are normally distributed: with mean 0 and variance

Confidence interval for 0 using b 0 Sample estimate ± margin of error

Hypothesis testing for 0 using b 0 The null (H 0 : 0 = ) versus the alternative (H A : 0 ≠ ) Test statistic P-value = How likely is it that we’d get a test statistic t* as extreme as we did if the null hypothesis is true? The P-value is determined by comparing the test statistic t* to a t-distribution with n-2 degrees of freedom.

Example 1: Inference The regression equation is punt = leg Predictor Coef SE Coef T P Constant leg S = R-Sq = 62.7% R-Sq(adj) = 59.3% Unusual Observations Obs leg punt Fit SE Fit Residual St Resid R R denotes an observation with a large standardized residual

Example 2: Inference The regression equation is Mortality = Latitude Predictor Coef SE Coef T P Constant Latitude S = R-Sq = 68.0% R-Sq(adj) = 67.3% Unusual Observations Obs Latitude Mortality Fit SE Fit Residual St Resid R X R R denotes an observation with a large standardized residual X denotes an observation whose X value gives it large influence.

Example 3: Inference The regression equation is consumption = dose Predictor Coef SE Coef T P Constant dose S = R-Sq = 73.9% R-Sq(adj) = 72.8% Unusual Observations Obs dose consumpt Fit SE Fit Residual St Resid R R denotes an observation with a large standardized residual.

Inference for 0 and 1 in Minitab Select Stat. Select Regression. Select Regression … Specify Response (y) and Predictor (x). Click on OK.