Confidence Intervals Help

When to be Skeptical

In some situations, the availability of statistical data can affect the very event the data is designed to analyze or predict. Cancer and hurricanes don't care about polls, but people do!

If you hear, for example, that there is a "95% chance that Dr. J will beat Mr. H in the next local mayoral race," take it with a dose of skepticism. There are inherent problems with this type of analysis, because people's reactions to the publication of predictive statistics can affect the actual event. If broadcast extensively by the local media, a statement suggesting that Dr. J has the election "already won" could cause overconfident supporters of Dr. J to stay home on election day, while those who favor Mr. H go to the polls in greater numbers than they would have if the data had not been broadcast. Or it might have the opposite effect, causing supporters of Mr. H to stay home because they believe they'll be wasting their time going to an election their candidate is almost certain to lose.

99.7% Confidence Interval

The empirical rule also states that in a normal distribution, 99.7% of the elements in a sample have a parameter that falls within three standard deviations of the mean for that parameter. From this fact we can develop an estimate of the 99.7% confidence interval.

In our situation, 99.7% of the bulbs can be expected to draw current that falls in a range equal to the estimate of the mean plus or minus three standard deviations (μ* ± 3σ*). In Fig. 5-10, this range is 2.910 amperes to 4.290 amperes.

c% Confidence Interval

We can obtain any confidence interval we want, within reason, from a distribution when we have good estimates of the mean and standard deviation (Fig. 5-11). The width of the confidence interval, specified as a percentage c, is related to the number of standard deviations x either side of the mean in a normal distribution. This relationship takes the form of a function of x versus c.

When graphed for values of c ranging upwards of 50%, the function of x versus c for a normal distribution looks like the curve shown in Fig. 5-12. The curve "blows up" at c = 100%.

Inexactness and Impossibility

The foregoing calculations are never exact. There are two reasons for this.

First, unless the population is small enough so we can test every single element, we can only get estimates of the mean and standard deviation, never the actual values. This can be overcome by using good experimental practice when we choose our sample frame and/or samples.

Second, when the estimate of the standard deviation σ* is a sizable fraction of the estimate of the mean μ*, we get into trouble if we stray too many multiples of σ* either side of μ*. This is especially true as the parameter decreases. If we wander too far to the left (below μ*), we get close to zero and might even stumble into negative territory – for example, "predicting" that we could end up with a light bulb that draws less than no current! Because of this, confidence interval calculations work only when the span of values is a small fraction of the estimate of the mean. This is true in the cases represented above and by Figs. 5-8, 5-9, and 5-10. If the distribution were much flatter, or if we wanted a much greater degree of certainty, we would not be able to specify such large confidence intervals without modifying the formulas.

Sampling and Estimation Practice Problems

Practice 1

Suppose you set up 100 archery targets, each target one meter (1 m) in radius, and have thousands of people shoot millions of arrows at these targets from 10m away. Each time a person shoots an arrow at the target, the radius r at which the arrow hits, measured to the nearest millimeter (mm) from the exact center point P of the bull's eye, is recorded and is fed into a computer. If an arrow hits to the left of a vertical line L through P, the radius is given a negative value; if an arrow hits to the right of L, the radius is given a positive value as shown in Fig. 5-13. As a result, you get a normal distribution in which the mean value, μ, of r is 0. Suppose you use a computer to plot this distribution, with r on the horizontal axis and the number of shots for each value of r (to the nearest millimeter) on the vertical axis. Suppose you run a program to evaluate this curve, and discover that the standard deviation, σ, of the distribution is 150 mm. This is not just an estimate, but is an actual value, because you record all of the shots.

Fig. 5-13. Illustration for Practice 1.

If you take a person at random from the people who have taken part at the experiment and have him or her shoot a single arrow at a target from 10m away, what is the radius of the 68% confidence interval? The 95% confidence interval? The 99.7% confidence interval? What do these values mean? Assume there is no wind or other effect that would interfere with the experiment.

Solution 1

Practice 2

Draw a graph of the distribution resulting from the experiment described in Practice 1, showing the 68%, 95%, and 99.7% confidence intervals.

Solution 2

Practice problems for these concepts can be found at:

Sampling and Estimation Practice Test