A.6 confidence intervals
- 2. • Confidence interval: used to describe the amount of uncertainty associated with a sample estimate of a population parameter • How to interpret a CI: You try first: How would you interpret this: “There is a 95% CI that states that the population mean is less than 150 but greater than 75.” incorrect interpretation: there is a 95% chance that the population mean falls between 75 and 150 **since the population mean is a population parameter, the population mean is a constant, not a random variable Let’s take a look at what the confidence level means first before we talk about the correct interpretation Confidence Level: describes the uncertainty associated with a sampling method What does this mean? Suppose we use a sampling method to select different samples and compute the interval estimate for each sample. Then a 95% confidence level says that we should expect 95% of the interval estimates to include a population parameter and the same for an other confidence level.
- 3. Confidence Interval Data Requirements Need ALL three: • Confidence Level • Statistic • Margin of Error • With the above information, the range of the CI is defined by: sample statistic + margin of error • Uncertainty associated with the CI is specified by the CL If Margin of Error not given, must calculate: ME= Critical Value x Standard Deviation of Statistic OR ME= Critical Value x Standard Error of Statistic
- 4. How to Construct a Confidence Interval 1. Identify a sample statistic {choose the statistic that you will use to estimate the population parameter (sample mean, sample proportion, etc.)} 2. Select a confidence level (usually 90%, 95%, 99%) 3. Find Margin of Error (ME) 4. Specify Confidence Interval: Sample statistic + ME
- 9. ** where z* or t* can be found using the tables and represents the standard error
- 12. How confidence intervals behave Typically a person performing an observational study chooses the confidence he desires and the margin of error follows from this choice. We usually want high confidence and a small margin of error, but we cannot have both. There is usually a trade-off. • If we ask for high confidence, we have to allow ourselves a large margin of error. Example: If I want to predict your average in a course with 99% confidence, I might say that I am 99% confident that you will get a 75% with a margin of error of 25%. That is saying that we are 99% confident that you will get between 50% and 100%. Notice that this doesn’t say much other than you will probably pass the course. • If we want a small margin of error, we have to ask for a smaller confidence level. Example: If I want to predict your average with a margin of error of 2 points, I might say that I am 50% confident that you will get a 92% with a margin of error of 2 percentage points. That is saying that I am 50% confident that you will get between a 90 and 94 in the course. Again, the small range is impressive but with 50% confidence, I am not very confident at all. It is a coin flip.
- 14. Sample Size for Desired Margin of Error • Sometimes we wish to establish a specified margin of error for a certain confidence level. That fixes z* and σ certainly cannot change. The only way we can achieve what we want is to change n, the sample size.