Confidence Intervals INTRO

Confidence Intervals Brief review of sampling. Brief review of the Central Limit Theorem. How do CIs work? Why do we use CIs? How do we calculate CIs? – CI for a proportion – CI for a mean How do we interpret CIs?

Brief Review of Sampling A sample is a subset of the population It is a collection of elements from the population Population: ALL CHHS students (ALL 250 of them) Sample of CHHS students: A group of 25 CHHS students

Sampling A sample can be different from the population (in its composition), but if it was collected in an unbiased way it is still representative. It is very unlikely that a sample will look EXACTLY like the population because of sampling error.

Total population = 100 people = 65 (65 % of the population is orange) = 9 (9% of the population is green) = 22 (22% of the population is blue) = 4 (4% of the population is pink) Population of CHHS students Suppose I select this group of students randomly by picking their names out of a hat that contained the names of all the students in CHHS It is a representative sample ? It does not matter that the sample composition is somewhat different to the population (40% orange) it is still representative because elements were selected randomly so no bias was introduced in the sampling process. The difference between the population and a representative sample is due to SAMPLING ERROR Representative sample

Sampling If representative samples are not likely to look exactly like the population (because of sampling error) how on earth can I make inferences about the population???? Statistical theory allows you to do so..

How do confidence intervals work? Statistical theory tells us that (in general) when we take random samples from a population of mean the means of those samples will follow a normal distribution with mean And this is regardless of how the population was distributed. This is the power of the Central Limit Theorem! The most important result (in my opinion) in statistical theory μ μ

Central Limit Theorem μ x If we 1.Take many many samples of size n and calculate the (sample) mean of each of those samples (x) and their standard deviations (sd) and 2.We plotted those means. The plot would follow a normal distribution with a mean equal to the population mean μ

How do confidence intervals work? x Suppose we get a sample of size n and calculate the sample mean x and standard deviation sd. We don’t know how far exactly this sample mean is from the population, but we know that all means for the samples we take from that population are normally distributed so we can be certain that the population mean will fall within approximately 2 standard deviations of the sample mean 95% of the time We don’t know where the population mean is, but we know it will fall within this distance 95 % of the time

How do confidence intervals work? Area =.025 μ x

Summary Representative samples are not likely to look exactly like the population because of sampling error. Sampling error is expected and we use statistics to deal with it. Confidence intervals are a way to deal with sampling error so that we can make accurate inferences about a population from a sample (even in the event of sampling error) The Central Limit Theorem provides the theoretical foundation for confidence intervals It basically tells us that a sample will fall within about 2 standard deviations from the population parameter about 95% of the time. And within about 3 standard deviations away from the population parameter 99% of the time.