Tolerance Intervals Example 1: Asthma Inhaler
How often does an asthma inhaler deliver the right dose in a puff? Is it enough that it delivers about the right dose on average? Need to know about individual puffs Example 2: Mental Health Testing
It is known that the top 10% of scorers on the OQ-45 test undergoing a particular therapy are likely to deviate from the recovery track. A therapist gives his patient the test. How does he know if his patient’s score is in this 10% range? Example 3: Releasing Land around A Uranium Enrichment Plant for Public Use
A state would like to make some of the land surrounding a uranium enrichment plant available for public use. Is all of the land safe for public recreational use?
How Do We Determine The Appropriate Cut-off
Example 1: FDA sets quality criteria for drug administration 85% coverage with high confidence If the inhaler cannot meet criteria it cannot be sold Are the points that mark the middle 85% of the data sufficient? Example 2: Need to identify 10% that are at risk Is the 90th percentile for test scores sufficient? Example 3: EPA insists that at least 95% of the land is within a safe range for uranium and uranium isotopes Soil samples collected from the area and analyzed for uranium Can the 95th percentile be used as the cut-off? Problem with Raw Percentiles
Would you make a decision about the mean of a drug based solely on the sample mean? Know that ̅ is not accurate Accuracy is dependent on sample size How do you know when the mean is too far away from a target to be due to random chance? Use hypothesis tests to account for this uncertainty Can use confidence intervals to perform a hypothesis test Allows a quick comparison between the critical value and sample mean What is a Confidence Interval?
Interval that we are (1-α)% confident covers μ Interval gets smaller as n increases Provides a range for practical interpretation of the mean Tells us nothing about individuals in the population Will A Confidence Interval Work?
All three examples require knowledge about individuals, not averages Example 1: Each puff matters, not the average puff Example 2: Need to know if my patient is at risk, not the average of everyone at risk Example 3: How do I know if all, or at least 95%, of the land is safe? Questions all deal with individuals, not averages Confidence intervals are not appropriate Tolerance Interval
Like a confidence interval for individuals Can cover a certain proportion of the population with a certain degree of confidence Example: a 99%/95% tolerance interval will include 99% of the population with 95% confidence. Confidence means the same thing it does with confidence intervals With a 90%/95% tolerance interval we want to capture 90% of the population with 95% confidence.
If this was the population only these two guys would be left out with 95% confidence.
In example 2 these are the two guys who are likely to deviate from the recovery track Basic Calculation
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