#### Transcript lecture29

Reviewing Confidence Intervals Anatomy of a confidence level • A confidence level always consists of two pieces: • A statistic being measured • A margin of error X m 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 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? Solution… • Use the correct z-value for 90% 5% of area left of this point 95% 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% X m 0.15 mg (1.645)0.025 mg 0.15 0.041mg 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: z* X z * X z * , z* 1.645 0.15 1.645 0.025 X 0.15 1.645 0.025 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: X z* n X z* Number of samples or tests We measure this We infer this n 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 = 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? Solution: • Since C = 0.90, z* = 1.645 z * n X z * n 17.5 1.645(9.3 ) X 17.5 1.645(9.3 ) 10 10 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? Margin of Error • When testing confidence limits you are saying that your statistical measure of the mean is: estimate +/- the margin of error • ie: X = 3.2 cm +/- 1.1 cm with a 90% confidence Math view… • Mathematically the margin of error is: z * n • 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: 1 1.4 2 1.7 3 1.5 4 1.9 5 1.8 Provide a 90% confidence level for the mean lifetime of these boards. 1.4 1.7 1.5 1.9 1.8 X 1.66 5 X z * 0.5 1.66 1.645( ) 1.66 0.37 n 5 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? m z* n n (z * m )2 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