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# Inference for Regression - PowerPoint PPT Presentation

Inference for Regression. Chapter 14. The Regression Model. The LSRL equation is ŷ = a + bx a and b are statistics; they are computed from sample data, therefore, we use them to estimate the true y-intercept, α , and true slope, β μ y = α + β x

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### Inference for Regression

Chapter 14

The LSRL equation is

ŷ = a + bx

a and b are statistics; they are computed from sample data, therefore, we use them to estimate the true y-intercept, α, and true slope, β

μy = α + βx

a and b from the LSRL are unbiased estimators of the parameters α and β

The null hypothesis

H0 : β = 0 A slope of 0 means (horizontal line) no correlation between x and y. The mean of y does not change at all when x changes.

The alternative hypothesis

Ha: β≠ 0 or Ha: β < 0 or Ha: β > 0

Negative slope Positive slope

When testing the hypothesis of no linear relationship a t statistic is calculated

In fact, the t statistic is just the standardized version of the least squares regression slope b.

t =

so we use table C to look up t and find the p-value.

The P-value is still interpreted the same way.

b is the slope from the least squares regression line, SEb is the standard

error of the least-squares slope b.

SEb =

Where

s =

And

SEb =

Therefore

t = or t =

t =

σ, the standard error about the LSRL (about y) is estimated by

s =

s =

• We have n observations of an explanatory variable x, and a

• response variable y. Our goal is to predict the

• behavior of y for given values of x.

• For any fixed value x, the response y, varies

• according to a normal distribution.

• Repeated responses of y are independent of

• each other.

• The mean response μy has a straight line relationship with x.

• μy = α + βx

• α and β are unknown parameters

• The standard deviation of y (call it σ) is the same for all values of x.

• (σ is unknown)

Steps forTesting Hypotheses of No Linear Relationship

• Make a scatter plot to make sure overall pattern of data is roughly linear (data should be spread uniformly above and below LSRL for all points)

• Make a plot of residuals because it magnifies any unusual pattern (again, uniform spread of data above and below y=0 line is needed)

• Make a histogram of the residuals to check that response values are normally distributed.

• Look for influential points that move the regression line and greatly can greatly affect the results of inference.

Minitab output always gives 2-sided p-value for Ha

If you want the p-value for alternative hypotheses of

Ha: β>0 or Ha: β< 0 just divide p-value from minitab by 2

βis the most important parameter in regression problem because it is the rate of change of the mean response as explanatory variable x, increases.

CI for β b ± t* Seb estimate± t* SEb

SEb = =

t * look up on table C with n-2 degrees of freedom

You can also find CI for αsame way, using SEa

a ± t*Sea (not commonly used)