STA220H5 Chapter Notes - Chapter 7.1: Normal Distribution, Central Limit Theorem, Confidence Interval
The unknown population parameter (e.g., mean or proportion) that we are
interested in estimating is called the target parameter.
Determining the target parameter
Parameter
Key words or phrases
Type of data
U
Mean; average
Quantitative
P
Proportion; percentage;
fraction; rate
Qualitative
A2 (optional)
Variance; variability;
spread
Quantitative
A point estimator of a population parameter is a rule or formula that tells us how to
use the sample data to calculate a single number that can be used as an estimate of
the target parameter.
An interval estimator (or confidence interval) is a formula that tells us how to use
the sample data to calculate an interval that estimates the target parameter.
The confidence coefficient is the probability that an interval estimator encloses the
population parameter-that is, the relative frequency with which the interval
estimator encloses the population parameter when the estimator is used repeatedly
a very large number of times. The confidence level is the confidence coefficient
expressed as a percentage.
The value za is defined as the value of the standard normal random variable z such
that the area a will lie to its right. In other words, P (z>za)=a
Large sample 100(1-a)% confidence interval for u, based on a normal (Z)
statistic
A known: x+-(za/2) ax=x+--(za/2)(a/n)
A unknown: x +- (za/2) ax=x+-(za/2)(s/n)
Where za/2 is the z-value corresponding to an area a/2 in the tail of a standard
normal distribution (see figure 7.5), ax is the standard deviation of the sampling
distribution of x, a is the standard deviation of the population, and s is the standard
deviation of the sample.
Conditions required for a valid large-sample confidence interval for u
1. A random sample is selected from the target population.
2. The sample size n is large (i.e., n>30). (due to the central limit theorem, this
condition guarantees that the sampling distribution of x is approximately
normal. Also, for large n, s will be a good estimator of a.)
Interpretation of a confidence interval for a population mean
When we form a 100(1-a)% confidence interval for u, we usually express our
confidence in the interval with a statement such as we can be (-a)% confident
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
The unknown population parameter (e. g. , mean or proportion) that we are interested in estimating is called the target parameter. A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter. An interval estimator (or confidence interval) is a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter. The confidence coefficient is the probability that an interval estimator encloses the population parameter-that is, the relative frequency with which the interval estimator encloses the population parameter when the estimator is used repeatedly a very large number of times. The confidence level is the confidence coefficient expressed as a percentage. The value za is defined as the value of the standard normal random variable z such that the area a will lie to its right.
