Confidence Intervals Help (page 3)

By — McGraw-Hill Professional
Updated on Sep 12, 2011

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.

Inexactness and Impossibility

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

The radius of the 68% confidence interval is equal to σ, or 150 mm. This means that we can be 68% confident that our subject's shot will land within 150mm of the center point, P. The radius of the 95% confidence interval is 2σ, or 300 mm. This means we can be 95% sure that the arrow will land within 300mm of P. The 99.7% confidence interval is equal to 3σ, or 450 mm. This means we can be 99.7% sure that the arrow will land within 450mm of P.

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

You should get a curve that looks like the one shown in Fig. 5-14.

Inexactness and Impossibility

Fig. 5-14. Illustration for Solution 2.

Practice problems for these concepts can be found at:

Sampling and Estimation Practice Test

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