Confidence intervals, t tests, P values

[Pages:42]Confidence intervals, t tests, P values

Joe Felsenstein Department of Genome Sciences and Department of Biology

Confidence intervals, t tests, P values ? p.1/31

Normality

Everybody believes in the normal approximation, the experimenters

because they think it is a mathematical theorem, the mathematicians

because they think it is an experimental fact!

G. Lippman

We can use the Gaussian (normal) distribution, assumed correct, and estimate the mean (which is the expectation). It turns out that, not surprisingly, the best estimate of the mean is the mean of the sample.

Confidence intervals, t tests, P values ? p.2/31

Normality

Everybody believes in the normal approximation, the experimenters

because they think it is a mathematical theorem, the mathematicians

because they think it is an experimental fact!

G. Lippman

We can use the Gaussian (normal) distribution, assumed correct, and estimate the mean (which is the expectation). It turns out that, not surprisingly, the best estimate of the mean is the mean of the sample.

(The median of the sample is a legitimate estimate too, but it is noisier).

Confidence intervals, t tests, P values ? p.2/31

Normality

Everybody believes in the normal approximation, the experimenters

because they think it is a mathematical theorem, the mathematicians

because they think it is an experimental fact!

G. Lippman

We can use the Gaussian (normal) distribution, assumed correct, and estimate the mean (which is the expectation). It turns out that, not surprisingly, the best estimate of the mean is the mean of the sample.

(The median of the sample is a legitimate estimate too, but it is noisier).

The sample mean is the optimal estimate as it is the Maximum Likelihood Estimate ? for which see later in the course.

Confidence intervals, t tests, P values ? p.2/31

Normality

Everybody believes in the normal approximation, the experimenters

because they think it is a mathematical theorem, the mathematicians

because they think it is an experimental fact!

G. Lippman

We can use the Gaussian (normal) distribution, assumed correct, and estimate the mean (which is the expectation). It turns out that, not surprisingly, the best estimate of the mean is the mean of the sample.

(The median of the sample is a legitimate estimate too, but it is noisier).

The sample mean is the optimal estimate as it is the Maximum Likelihood Estimate ? for which see later in the course. But how do we figure out how noisy the estimate is?

Confidence intervals, t tests, P values ? p.2/31

Normality

Everybody believes in the normal approximation, the experimenters

because they think it is a mathematical theorem, the mathematicians

because they think it is an experimental fact!

G. Lippman

We can use the Gaussian (normal) distribution, assumed correct, and estimate the mean (which is the expectation). It turns out that, not surprisingly, the best estimate of the mean is the mean of the sample.

(The median of the sample is a legitimate estimate too, but it is noisier).

The sample mean is the optimal estimate as it is the Maximum Likelihood Estimate ? for which see later in the course. But how do we figure out how noisy the estimate is?

Can we make an interval estimate?

Confidence intervals, t tests, P values ? p.2/31

A normal distribution (artist's conception)

how often

the true mean (expectation) true distribution

the measurement

Confidence intervals, t tests, P values ? p.3/31

Uncertainty of the mean

Let's go forward (distribution to data).

Confidence intervals, t tests, P values ? p.4/31

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