T-test.

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Inference in the Simple Regression Model

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Presentation transcript:

T-test

t A unitless number with a known distribution, if the assumptions about the errors are true. The Y values are random variables. You calculate the least squares slope from the Y values. Therefore, the slope estimate is a random variable.

t The slope has a mean and a variance. We can calculate those, based on the assumptions about the errors. The mean of the slope is the true slope. That’s what “unbiased” implies.

t The slope has a mean and a variance. We can calculate those, based on the assumptions about the errors. The mean of the slope is the true slope. That’s what “unbiased” implies.

Standard error of beta-hat

T-test This has the t-distribution with N-2 degrees of freedom. (The beta should be beta-0, your hypothesized value.)

t For testing the hypothesis that the true beta is 0: N-2 degrees of freedom.

Types of errors Type I error: Type II error: Rejecting a hypothesis that is true Type II error: Refusing to reject a hypothesis that is false. The significance level is the probability of a Type I error.

T table

Next time: Graphs How to tell if the assumptions are plausible. NOT by standard regression results.

Confidence interval for a coefficient Coefficient ± its standard error × t from table One calculation (two, really) lets you test many hypothesized values for the true parameter. If 0 is in the confidence interval, your coefficient is not significantly different from 0.

Confidence interval for a coefficient Coefficient ± its standard error × t from table 95% probability that the true coefficient is in the 95% confidence interval? If you do a lot of studies, you can expect that, for 95% of them, the true coefficient will be in the 95% confidence interval.

Confidence interval for prediction Hyperbolic outline