Statistical analysis Using R and Excel (SURE)
Chapter 07 Interval Estimation
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
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
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.
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 Confdence 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 .