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Reviewing Confidence Intervals

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:

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Reviewing Confidence Intervals

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  1. Reviewing Confidence Intervals

  2. 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

  3. Simple example • Suppose that we know the standard deviation for the active ingredient in a drug is 0.025 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?

  4. 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 -1.645 and +1.645 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%

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

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

  7. 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

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

  9. 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 m3 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 s = 9.3 fish with a sample mean of 17.5 fish. How precisely does he know the true mean of fish/6000 m3? Use C = 90% If the volume of the lake is 60 million m3, how many trout are in the lake?

  10. Solution: • Since C = 0.90, z* = 1.645 There is a 90% chance that the true mean number of fish/6000 m3 lies in the range (16.0,19.0) Total number of fish: He is 90% confident that there are between 160 000 and 190 000 fish in the lake. Why should you be skeptical of this result?

  11. 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

  12. 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 s smaller)

  13. 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 s = 0.5: Provide a 90% confidence level for the mean lifetime of these boards.

  14. IT is 90% confident that the mean lifetime of these boards is between 1290 and 2030 hours. 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!

  15. 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

  16. 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

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