Reviewing Confidence Intervals

Anatomy of a confidence level A confidence level always consists of two pieces: A statistic being measured A margin of error The margin of error can be determined by many different methods depending on what kind of distribution we are using: normal, t-test, paired tests etc Go to applet that demonstrates the concept of a confidence level

Simple example Suppose that we know the standard deviation for the active ingredient in a drug is mg and the variation in amount is normally distributed. If we measure a sample of the drug and find the amount of active ingredient present is 0.15 mg, what would be the acceptable range of active ingredient at the 90% confidence level?

Solution… Use the correct z-value for 90% 95% of area left of this point 5% of area left of this point The correct z values are and and are usually denoted z* to indicate that these are special ones chosen with a particluar confidence level “C” in mind. In this example C = 90%

Another way to express this is: The amount of active ingredient is (0.109,0.191) mg at the 90% level

Using the z-score formula we get: 90% of the readings will be expected to fall in the range ( 0.109,0.191 ) mg

Using Confidence Intervals when Determining the True value of a Population Mean We rarely ever know the population mean – instead we can construct SRS’s and measure sample means. A confidence interval gives us a measure of how precisely we know the underlying population mean We assume 3 things: We can construct “n” SRS’s The underlying population of sample means is Normal We know the standard deviation

This gives … Confidence interval for a population mean : We measure this We infer this Number of samples or tests

Example: Fish or Cut Bait? A biologist is trying to determine how many rainbow trout are in an interior BC lake. To do this he uses a large net that filters 6000 m 3 of lake water in each trial. He drops the net in a specific area and records the mean number of fish caught in 10 trials. This represents one SRS. From this he is able to determine a mean and standard deviation for the number of fish in 100 SRS’s. Each SRS has the same  = 9.3 fish with a sample mean of 17.5 fish. How precisely does he know the true mean of fish/6000 m 3 ? Use C = 90% If the volume of the lake is 60 million m3, how many trout are in the lake?

Solution: Since C = 0.90, z* = There is a 90% chance that the true mean number of fish/6000 m 3 lies in the range (16.0,19.0) Total number of fish: He is 90% confident that there are between and fish in the lake. Why should you be skeptical of this result?

Margin of Error When testing confidence limits you are saying that your statistical measure of the mean is: ie: X = 3.2 cm +/- 1.1 cm with a 90% confidence estimate +/- the margin of error

Math view… Mathematically the margin of error is: You can reduce the margin of error by increasing the number of samples you test making more precise measurements (makes  smaller)

Matching Sample Size to Margin of Error An IT department in a large company is testing the failure rate of a new high-end graphics card in 200 of its work stations. 5 cards were chosen at random with the following lifetime per failure (measured in 1000’s of hours) and  = 0.5: Provide a 90% confidence level for the mean lifetime of these boards.

IT is 90% confident that the mean lifetime of these boards is between 1290 and 2030 hours. However However – these are expensive boards and accounting wants to have the margin of error reduced to 0.10 with a 90% confidence level. What should IT do? IT needs to test 68 machines!

Using other statistical tests… The margin of error can be estimated in many different ways… Consider 7.37 Here we are using a confidence iterval to test the likelihood of the null hypothesis

The main idea… Margin of error shows you the range in a confidence interval The value of ME depends on the confidence level you set and the type of statistical analysis that is appropriate