Regression

Height Weight How much would an adult female weigh if she were 5 feet tall? She could weigh varying amounts – in other words, there is a distribution of weights for adult females who are 5 feet tall. This distribution is normally distributed. (we hope) What would you expect for other heights? 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 The mean response  y has a straight-line relationship with x: –Where: slope  and intercept  are unknown parameters For any fixed value of x, the response y varies according to a normal distribution. Repeated responses of y are independent of each other. The standard deviation of y (  y ) is the same for all values of x. (  y is also an unknown parameter)

The slope b of the LSRL is an unbiased estimator of the true slope . The intercept a of the LSRL is an unbiased estimator of the true intercept . The standard error s is an unbiased estimator of the true standard deviation of y (  y ). Note: df = n-2 We use to estimate

Linear Regression: Inference on the slope of a line Write justifications: 3) Independent random sample for the population being studied. 1) Residuals are normally distributed. 2) The y values at any given x value are approx. normal and all have the same standard deviation. Look at the normal probability plot of the of the residuals. Look at the scatter plot. Is the scatter plot relatively straight? State the parameter – β

Linear Regression: inference on the slope of a line df = n-2 Confidence Interval

Linear Regression: Inference on the slope of a line Write justifications: 3) Independent random sample for the population being studied. 1) Residuals are normally distributed. 2) The y values at any given x value are approx. normal and all have the same standard deviation. Look at the normal quartile plot of the of the residuals. Look at the scatter plot. Is the scatter plot relatively straight?

Linear Regression: Inference on the slope of a line Null Hypothesis: Alternative Hypothesis There is no change - slope is zero– there is not a useful linear relationship There is a change - slope is not zero– there is a useful linear relationship Test Statistic df = n-2 Reject H o – there is a useful linear relationship between x and y Fail to Reject H o – there is not a useful linear relationship between x and y is the standard error of the slope s is our sample estimate of the common standard deviation of errors around the true line Inference Test

The Leaning Tower of Pisa leans more as time passes. Here are measurements of the lean of the tower for the years 1975 to The lean is the distance between where a point on the tower would be if the tower were straight and where it actually is. The distances are tenths of a millimeter in excess of 2.9 meters. For example, the 1975 lean, which was meters, appears in the table as 642 Year Lean Does there appear to be clear evidence there is a strong linear relationship between year and lean?  = the true mean change in lean per year for the Leaning Tower of Pisa H 0 :  = 0 H a :   0

Conditions: 1) The residuals should be normally distributed. The normal probability plot of the residuals is fairly linear, so this condition is met. 2) The y-values at any given x value should be approximately normal and have the same standard deviation. The scatter plot is fairly linear so this condition is met. 3) The sample should be random and independent which we will assume. Since the conditions are met a test for slope is appropriate.

p-value = df = 11 Decision: Since p-value < , I will reject the null hypothesis at the.05 significance level. Conclusion: There is enough evidence to conclude that there is a useful linear relationship between year and lean of the Leaning Tower of Pisa.

The Leaning Tower of Pisa leans more as time passes. Here are measurements of the lean of the tower for the years 1975 to The lean is the distance between where a point on the tower would be if the tower were straight and where it actually is. The distances are tenths of a millimeter in excess of 2.9 meters. For example, the 1975 lean, which was meters, appears in the table as 642 Year Lean Construct a 95% confidence interval for the true slope of the regression line. df = 11 I am 95% confident, the true mean change in lean per year is between and tenths of a millimeter in excess of 2.9 meters per year, based on this regression

The management of a chain of package delivery stores would like to predict the weekly sales (in $1000) for individual stores based on the number of customers. Minitab output is for sample data is below. PredictorCoef Stdev t-ratio p Constant CUSTOMER s = R-sq = 91.2%R-sq (adj) = 90.7% Analysis of Variance Source DF SS MS F P Regression 1SSR Error 18 SSE Total 19 SST The regression equation is a bSE b t df p-value