Regression Inference

What would you expect for other heights? Weight What would you expect for other heights? How much would an adult male weigh if he were 5 feet tall? He could weigh varying amounts (in other words, there is a distribution of weights for adult males who are 5 feet tall) This distribution is normal (we hope) Where would you expect the TRUE LSRL to be? What about the standard deviations of all these normal distributions? We want the standard deviations of all these normal distributions to be the same.

Regression Model x & y have a linear relationship with the true LSRL going through each my sy is the same for each x-value For a given x-value, the responses (y) are normally distributed

Regression Model For each value of x, y is a normal distribution  a and b are unknown parameters For each value of x, y is a normal distribution Standard deviation of y is the same for all values of x To estimate, , we use Unbiased estimator of sy (true SD of y) : s (SD of the residuals) df = n – 2, since there are 2 unknowns

It is difficult to accurately determine a person’s body fat percentage without immersing him or her in water. Researchers hoping to find ways to make a good estimate immersed 20 male subjects, then measured their weights. Find the LSRL, correlation coefficient, and coefficient of determination. Body fat = -27.376 + 0.250(weight) r = 0.697 r2 = 0.485

b) Interpret the slope in this context. For every increase of a pound in weight, there is an approximate .25% increase in body fat. c) Interpret the coefficient of determination in this context. Approximately 48.5% of the variation in body fat can be explained by the LSRL of body fat and weight.

e) Create a scatter plot and residual plot for the data. d) Estimate α, β, and σy. α ≈ -27.376 β ≈ 0.25 σy ≈ 7.049 e) Create a scatter plot and residual plot for the data. Weight Body fat Weight Residuals

Sampling Distribution of Slopes

Regression Model x & y have a linear relationship with the true LSRL going through each my sy is the same for each x-value For a given x-value, the responses (y) are normally distributed

Height Weight Suppose you took many samples of the same size from this population & calculated the LSRL for each. Using the slope from each of these LSRLs, we can create a sampling distribution for the slope of the true LSRL. What is the standard deviation of the sampling distribution? What will the mean of the sampling distribution equal? What shape will this distribution have? b b b b b b b mb = b

What is the slope of a horizontal line? Height Weight Suppose the LSRL is a horizontal line. Would height be useful in predicting weight? A slope of zero means there is NO relationship between x & y!

Hypotheses Define b in context! H0: b = 0 Ha: b > 0 Ha: b < 0 There is no relationship between x & y (x should not be used to predict y) H0: b = 0 Ha: b > 0 Ha: b < 0 Ha: b ≠ 0 Define b in context!

Formulas: Confidence Interval: Hypothesis Test: df = n – 2

f) Is there evidence that weight can be used to predict body fat? H0: β = 0 Where β is the true slope of the LSRL of weight Ha: β ≠ 0 & body fat Since p-value < α, we reject H0. There is sufficient evidence to suggest that weight can be used to predict body fat. p-value = .0006, α = .05

g) Construct a 95% confidence interval for the true slope of the LSRL. We are 95% confident that the true slope of the LSRL of weight & body fat is between 0.12 and 0.38. If we made lots of intervals this way, 95% of them would contain the true slope.

Conditions for Inference on Slope SRS (unbiased) No pattern in residuals (linear) Points evenly spaced across LSRL (constant sy) Residuals graph shows normality