Use R to analytical data(I have presentation file)

Statistical analysis Using R and Excel (SURE)

Chapter 07 Interval Estimation

( ① A point estimator of a population parameter is a rule that tells you how to calculate a single number based on sample data . ② An interval estimator of a population parameter is a rule that tells you how to calculate two numbers (an interval) based on sample data. )

1. Definitions

- When an interval estimator is employed to estimate a population parameter, the pair of numbers obtained from the estimator is called an interval estimate or confidence interval for the parameter. The larger number, which locates the upper end of the interval, is called the upper confidence limit and is denoted by UCL, the number that locates the lower extreme of the interval is called the lower confidence limit and is denoted by LCL.

- An estimator of a population parameter is said to be unbiased if the mean of its sampling distribution is equal to the parameter. Otherwise, the estimator is said to be biased.

- The distance between an estimate and the estimated parameter is called the error of estimation.

- The probability that a confidence interval will enclose the estimated parameter is called the confidence coefficient..

2. A Confidence Interval for the Population Mean μ

1) σ known case

- → )

- .

- 100(1-α)% Confidence Interval for μ:

- the margin of error:

ex) 1-α = 0.95 →

2) σ unknown case

- t = follows t-distribution with degree of freedom n with E(t) = 0, Var(t) > 1.

- The t distribution is a family of similar probability distributions, with a specific t distribution depending on a parameter known as the degrees of freedom.

- 100(1-α)% Confidence Interval for μ:

where s is the sample standard deviation, (1-α) is the confidence coefficient, and is the t value that providing an area of α/2 in the upper tail of the t distribution with n-1 degrees of freedom.

3. A Confidence Interval for the Population Proportion p

① If we choose for an interval estimate for p, it cannot be used directly because p is what we are trying to estimate. So is substituted for p and the margin of error for an interval estimate of a population proportion is given by

Margin of error =

② 100(1-α)% Confidence Interval for p

where (1-α) is the confidence coefficient and is the z value providing

an area of α/2 in the upper tail of the standard normal distribution.

4. Determining the Sample Size

1) Sample Size for an Interval Estimate of a Population Mean

where E is the desired margin of error.

2) Sample Size for an Interval Estimate of a Population Proportion

where E is the desired margin of error and p* is the planning value for .