Confidence Interval (CI)
- is the confidence in how accurate an estimated statistic is within range of the true parameter. Given a sample of a population, we analyze the sample and compute its statistics (e.g. sample mean, sample variance, etc). A statistic 𝜃ˆ is an estimate of the true parameter 𝜃 of the ENTIRE population. Since 𝜃ˆ are computed from a random sample they are not likely to be equal to the true parameter. This is the risk of sampling which is taking a sample and making generalizations of the larger population. Confidence Intervals allow us to measure that risk.
- A point estimate/sample statistic 𝜃ˆ is an estimate of the unknown parameter 𝜃 of the population. We know that it is likely NOT exactly
- 𝜃ˆ = 𝜃 # This proposition is not likely to be true
- How much trust can we then put into 𝜃ˆ? We can use Confidence Intervals (CI)
CI - Definition
an interval [𝐴, 𝐵] is a (1 − 𝛼)100% confidence interval for the parameter 𝜃 if it contains the parameter with probability (1 − 𝛼):
- 𝐏{𝐴 ≤ 𝜃 ≤ 𝐵} = 1 − 𝛼
where:
- 𝛼 - significance level
- (1 − 𝛼) - confidence level or coverage probability
CI - What is It?
Click here to expand...
the statement:
“There is a 95% CHANCE that the sample mean 𝑋̅ falls within 3 units of population mean 𝜇”
can be rephrased as:
“We are 95% CONFIDENT that the population mean 𝜇 falls within 3 units of the 𝑋̅ we found in our sample.”
Note that the first phrasing is about 𝑋̅, which is a random variable; that’s why it makes sense to use probability language. But the second phrasing is about 𝜇, which is a parameter, and thus is a “fixed” value that does not change, and that’s why we should not use probability language to discuss it. In these problems, it is our 𝑋̅ that will change when we repeat the process, not 𝜇
Example
We are 95% confident that the mean SAT-M score of all community college students in the researcher’s state is covered by the interval (467.3, 482.7). Note that the confidence interval was obtained by taking 475 ± 7.7. This means that we are 95% confident that by using the sample mean (𝑋̅ = 475) to estimate 𝜇, our error is no more than 7.7 points
There is a trade-off between the level of confidence and the precision with which the parameter is estimated
the wider 99% confidence interval (111, 119) gives us a less precise estimation about the value of 𝜇 than the narrower 90% confidence interval (112.5, 117.5), because the smaller interval ‘narrows-in’ on the plausible values of 𝜇
CI - Formula For Unbiased Estimator 𝜃ˆ With Normal Distribution
CI - In Specific Problems
- Confidence Intervals in 1 Sample Problems - inference on the parameter(s) of a single population
- Confidence Intervals in 2 Sample Problems - inference on parameters involving two populations