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T-test

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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.

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**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.

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**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.

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**Standard error of beta-hat**

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

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**t For testing the hypothesis that the true beta is 0:**

N-2 degrees of freedom.

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**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.

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T table

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**Next time: Graphs How to tell if the assumptions are plausible.**

NOT by standard regression results.

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**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.

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**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.

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**Confidence interval for prediction**

Hyperbolic outline

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