The approximate intervals are calculated using the Z-Distribution (standard normal distribution). The exact intervals are calculated using the T-Distribution.

Confidence and Prediction Interval of Mean

What is known?

Mean

SD

N

Approx:

SE

% PI

% CI

Null Hypothesis (H0)

Z Value

P value

Mean

Exact:

SE

% PI

% CI

Null Hypothesis (H0)

T Value

P value

Mean

Confidence Interval of Proportion

What is known?

Events (cases)

N

Approx:

Proportion

SE

% CI

Null Hypothesis (H0)

Z Value

P value

Proportion

Exact:

Proportion

SE

% CI

Null Hypothesis (H0)

T Value

P value

Proportion

Confidence Interval of Slope (Regression Coefficient)

The slope of a line is in this case the β value in the expression \( Y = \beta X + \alpha \: (+error)\) from a linear regression.
N is the number of points (x, y). The degrees of freedom is DF = N - 2.
The correlation coefficient R^{2} is defined by \(R^2 = \frac{SSR}{SST}\) or \(R^2 = 1 - \frac{SSE}{SST}\)

SST = Sum of the Squares (Total) = SSE + SSR
SSR = Sum of the Squares (due to Regression)
SSE = Sum of Squares (due to Error)