Chapter 6 Wrap Up
Concept Check
Section Reviews
6.1 Point Estimation and Sampling Distributions
Since Populations are typically large and a census may not be feasible, we often use sample statistics to estimate population parameters. Some examples of point estimates are:
- is a point estimate for μ
- p̂ is a point estimate for ρ
- s is a point estimate for σ
However, we know sampling variability exists, so each statistic has it’s own probability distribution called a sampling distribution. In order for the statistic to be unbiased, the center of this sampling distribution should be equal to the parameter of interest (accurate), and the standard error tells us about the precision of the estimate.
6.2 Sampling Distribution of the Sample Mean
In a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (n).
The Central Limit Theorem for Sample Means: ~ N
The Mean : μx
Central Limit Theorem for Sample Means z-score and standard error of the mean:
Standard Error of the Mean (Standard Deviation ():
6.3 Intro to Confidence Intervals
In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. A confidence interval is made up of the point estimate with a Margin of Error built in (MoE) A CI has the general form:
(lower bound, upper bound) = (point estimate – MoE, point estimate + MoE)
The calculation of the MoE depends on the size of the sample and the level of confidence desired. The confidence level is the percent of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding MoE increases as well. As the sample size increases, the MoE decreases. By the central limit theorem,
Given a confidence interval, you can work backwards to find the error bound (MoE) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.
CL = confidence level, or the proportion of confidence intervals created that are expected to contain the true population parameter
α = 1 – CL = the proportion of confidence intervals that will not contain the population parameter
For a Single Population Mean with known Standard Deviation we can use the N