.:INTRODUCTORY STATISTICS

: INTERVAL ESTIMATION

Outline2

Interval estimation: motivation

What is interval estimation?

Formulas for confidence interval

Calculation of sample size

Interval estimation: motivation I3

Population parameters (such as the population mean μ and thepopulation proportion p) are impractical to calculate because it isusually expensive to conduct a census.

But recall that type of statistics called inferential statistics. Thatpart of statistics involves using sample data to estimate (or toinfer) population parameters.

Sample statistics such as the sample mean X and the sampleproportion ˆ , calculated from sample data, are used as point

estimates of the population parameters μ and p, respectively.

A point estimate (aka sample statistics) is one value used toapproximate a population parameter.

Interval estimation: motivation II4

However, point estimates (or sample statistics) are alsoproblematic because they vary from one sample to the next.

Since we are stuck with selecting one sample, it is likely that thepoint estimate calculated from that one sample may be quitedifferent from the population parameter it is trying to estimate.Therefore, it is not prudent to use a point estimate, by itself.

We must now find a more prudent and reliable method ofestimating population parameters.

A more prudent and reliable method of estimation is intervalestimation.

What is interval estimation?5

Interval estimation is a technique used to estimate aconfidence interval, typically, for the unknown populationmean (μ) and proportion ( p ).

A confidence interval gives an estimated (from sample data)range of values which is likely to include the unknownpopulation parameter with some level of confidence.

The confidence level is a measure of how confident one isthat the confidence interval contains the populationparameter.

A confidence interval has a lower limit and an upper limitcalled confidence limits.

Formulas for confidence interval I

6

The confidence interval for thepopulation mean is

Margin of error, e

Z is a value that is based on the levelof confidence.

Interpretation: we are ## %confident that the interval contains thetrue population mean.

is called the standard error for the sample mean.

The confidence interval for the population proportion is

Margin of error, e

Interpretation: we are ## %confident that the interval contains thetrue population mean.

Z is a value that is based on the levelof confidence.

is called the standard error for the sample mean.

Formulas for confidence interval II7

Z, which is used in the formula for confidence interval, is called the critical value. The value of Z depends on the confidence level.

The following are typical values of Z based on typical confidence levels.

Confidence level Z-value90% 1.644995% 1.9698% 2.326399% 2.5758

Example I8

1.

2.

3.

Calculation of sample size9

When carrying out a sample survey, a sample must be selected froma population. But, how large a sample should be used?

The sample size typically depends on the level of confidence, thestandard deviation and the margin of error.

If σ is known, the sample size based on the mean is computed as:

The sample size based on the proportion is calculated as:

Example II10

1.

2.