Chapter 7: Confidence Intervals for One Population

Statistical inference is used to draw conclusions about a population based on a sample. We can use the probability distributions and Central Limit Theorem to understand what is going on in the population. The population can be difficult to measure so we take a sample from that population and use descriptive statistics to measure the sample. We can then use those sample statistics to infer back to what is happening in our population. Although there are many types of statistical inference tools, we will only cover some of the more common techniques.

Distinguishing between a population and a sample is important in statistics. We frequently use a representative sample to generalize a population.

• A statistic is any characteristic or measure from a sample. One example is the sample mean $\overline{x}$.
• A parameter is any characteristic or measure from a population. One example is the population mean µ.
• A point estimate for a parameter (a characteristic from a population) is a statistic (a characteristic from a sample). For example, the point estimate for the population mean µ is the sample mean $\overline{x}$. The point estimate for the population standard deviation σ is the sample standard deviation s, etc.
• A 100(1 – α)% confidence interval for a population parameter (μ, σ, etc.) represents that the proportion 100(1 – α)% of times the true value of the population parameter is contained within the interval.
• The confidence level (or level of confidence) is 1 – α. The common percentages used for confidence interval levels are 90%, 95%, and 99%. Some corresponding values of alpha are: 90% would be α = 0.10 = 10%, 95% would be α = 0.05 = 5%, and 99% would be α = 0.01 = 1%. In this context, α, “alpha,” represents the complement of the confidence level, and its definition will be explained in the next chapter.

When a symmetric distribution, such as a normal distribution, is used, confidence intervals are always of the form: point estimate ± margin of error