Confidence Intervals Help
Introduction to Confident Intervals
A distribution can provide us with generalized data about populations, but it doesn't tell us much about any of the individuals in the population. Confidence intervals give us a better clue as to what we can expect from individual elements taken at random from a population.
Suppose we live in a remote scientific research outpost where the electric generators only produce 90 volts. The reason for the low voltage is the fact that the generators are old, inefficient, and too small for the number of people assigned to the station. Funding has been slashed, so new generators can't be purchased. There are too many people and not enough energy. (Sound familiar?)
We find it necessary to keep the compound well-lit at night, regardless of whatever other sacrifices we have to make. So we need bright light bulbs. We have obtained data sheets for 550-watt sodium-vapor lamps designed for 120-volt circuits, and these sheets tell us how much current we can expect the lamps to draw at various voltages. Suppose we have obtained the graph in Fig. 5-8, so we have a good idea of how much current each bulb will draw from our generators that produce 90 volts. The estimate of the mean, μ*, is 3.600 amperes. There are some bulbs that draw a little more than 3.600 amperes, and there are some that draw a little less. A tiny proportion of the bulbs draw a lot more or less current than average.
If we pick a bulb at random, which is usually what happens when anybody buys a single item from a large inventory, how confident can we be that the current our lamp draws will be within a certain range either side of 3.600 amperes?
68% Confidence Interval
Imagine that our data sheets tell us the standard deviation of the distribution shown in Fig. 5-8 is 0.230 amperes. According to the empirical rule, which we learned about in Chapter 3, 68% of the elements in a sample have a parameter that falls within one standard deviation (± σ) of the mean μ for that parameter in a normal distribution. We don't know the actual standard deviation or the actual mean μ for the lamps in our situation, but we have estimates μ* and σ* that we can use to get a good approximation of a confidence interval.
In our situation, the parameter is the current drawn at 90 volts. Therefore, 68% of the bulbs can be expected to draw current that falls in a range equal to the estimate of the mean plus or minus one standard deviation (μ* ± σ*). In Fig. 5-8, this range is 3.370 amperes to 3.830 amperes. It is a 68% confidence interval because, if we select a single bulb, we can be 68% confident that it will draw current in the range between 3.370 and 3.830 amperes when we hook it up to our sputtering, antiquated 90-volt generator.
95% Confidence Interval
According to the empirical rule, 95% of the elements have a parameter that falls within two standard deviations of the mean for that parameter in a normal distribution. Again, we don't know the actual mean and standard deviation. We only have estimates of them, because the data is not based on tests of all the bulbs of this type that exist in the world. But we can use the estimates to get a good idea of the 95% confidence interval.
In our research-outpost scenario, 95% of the bulbs can be expected to draw current that falls in a range equal to the estimate of the mean plus or minus two standard deviations (μ* ± 2σ*). In Fig. 5-9, this range is 3.140 amperes to 4.060 amperes.
The 95% confidence interval is often quoted in real-world situations. You may hear that "there's a 95% chance that Ms. X will survive her case of cancer for more than one year," or that "there is a 95% probability that the eyewall of Hurricane Y will not strike Miami." If such confidence statements are based on historical data, we can regard them as reflections of truth. But the way we see them depends on where we are. If you have an inoperable malignant tumor, or if you live in Miami and are watching a hurricane prowling the Bahamas, you may take some issue with the use of the word 'confidence" when talking about your future. Statistical data can look a lot different to us when our own lives are on the line, as compared to the situation where we are in some laboratory measuring the currents drawn by light bulbs.
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