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45-733 PROBABILITY AND STATISTICS I

A SHORT GUIDE TO CONFIDENCE INTERVALS

Confidence Sample Size Variance What Test Interval (Type of Distribution) to Use

s^{2}known IA Z Small Sample (Normal) s^{2}unknown IIIA t m s^{2}known IA Z Large Sample (Any) s^{2}unknown IIA Z

s^{2}known IB Z Small Sample (Normal) s^{2}unknown IIIB t m_{1}- m_{2}s^{2}known IB Z Large Sample (Any) s^{2}unknown IIB Z

p Large Sample NA VA Z

p_{1}- p_{2}Large Sample NA VB Z

s^{2}Small and Large (Normal) NA IV c^{2}

- Confidence Intervals for
**m**When**s**is known^{2}

These tests are used when the random sample is drawn from a.*Normal distribution with a known variance*

- Confidence Interval for
**m**

Confidence Limits

- Confidence Interval for the Difference Between Two Means

In this sampling situation the variances of the two populations are known.

Confidence Limits

- Confidence Interval for
- Large Sample Confidence Intervals for
**m**(n ³ 30)

When the sample size is large, the Central Limit Theorem allows us to calculate confidence intervals forusing these formulas (compare these with the ones for proportions in*any distribution***V**).

- Confidence Interval for
**m**

where

Confidence Limits

- Confidence Interval for the Difference Between Two Means

Confidence Limits

- Confidence Interval for
- Small Sample Confidence Intervals for
**m**(n £ 29)

These tests are used when the random sample is drawn from a.*Normal distribution with unknown variance*

- Confidence Interval for
**m**where

Confidence Limits

- Confidence Interval for the Difference Between Two Means

In this sampling situation the variances of the two populations are assumed to be the same.

where

- Confidence Interval for
- Confidence Interval For
**s**^{2}

This test is used when the random sample is drawn from a.*Normal Distribution*

where

Confidence Limits and

- Large Sample Confidence Intervals For Proportions

When the sample size is large, the

allows us to calculate confidence intervals for proportions (technically the random sample is drawn from a*Central Limit Theorem*). Recall that*Bernoulli distribution*

~

- Confidence Interval for
**p**

Confidence Limits

- Confidence Interval For the Difference Between Two Proportions

Let

**p**and_{1}**p**be the two population proportions and let_{2}**n**and**m**be the sample sizes from the two populations respectively. Using thewe assume that*Central Limit Theorem*~

where

- Confidence Interval for