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268 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

EXERCISES FOR SECTION 8-5
8-42. Of 1000 randomly selected cases of lung cancer, 823 8-45. The Arizona Department of Transportation wishes to
resulted in death within 10 years. Construct a 95% two-sided survey state residents to determine what proportion of the
confidence interval on the death rate from lung cancer. population would like to increase statewide highway speed
8-43. How large a sample would be required in Exercise limits to 75 mph from 65 mph. How many residents do they
8-42 to be at least 95% confident that the error in estimating need to survey if they want to be at least 99% confident that
the 10-year death rate from lung cancer is less than 0.03? the sample proportion is within 0.05 of the true proportion?
8-44. A random sample of 50 suspension helmets used by 8-46. A manufacturer of electronic calculators is interested
motorcycle riders and automobile race-car drivers was sub- in estimating the fraction of defective units produced. A ran-
jected to an impact test, and on 18 of these helmets some dam- dom sample of 800 calculators contains 10 defectives.
age was observed. Compute a 99% upper-confidence bound on the fraction
(a) Find a 95% two-sided confidence interval on the true pro- defective.
portion of helmets of this type that would show damage 8-47. A study is to be conducted of the percentage of home-
from this test. owners who own at least two television sets. How large a
(b) Using the point estimate of p obtained from the prelimi- sample is required if we wish to be 99% confident that the
nary sample of 50 helmets, how many helmets must be error in estimating this quantity is less than 0.017?
tested to be 95% confident that the error in estimating the 8-48. The fraction of defective integrated circuits produced
true value of p is less than 0.02? in a photolithography process is being studied. A random sam-
(c) How large must the sample be if we wish to be at least ple of 300 circuits is tested, revealing 13 defectives. Find a
95% confident that the error in estimating p is less than 95% two-sided CI on the fraction of defective circuits pro-
0.02, regardless of the true value of p? duced by this particular tool.

8-6 A PREDICTION INTERVAL FOR A FUTURE OBSERVATION

In some problem situations, we may be interested in predicting a future observation of a
variable. This is a different problem than estimating the mean of that variable, so a confidence
interval is not appropriate. In this section we show how to obtain a 100(1 )% prediction
interval on a future value of a normal random variable.
Suppose that X , X , p , X is a random sample from a normal population. We wish to
n
1
2
predict the value X n 1 , a single future observation. A point prediction of X n 1 is X,
the sample mean. The prediction error is X n 1 X. The expected value of the prediction
error is
E 1X n 1 X 2 0

and the variance of the prediction error is

2 1
2
2
V 1X n 1 X 2 n a1 b
n
X
because the future observation, X n 1 is independent of the mean of the current sample . The
prediction error X n 1 X is normally distributed. Therefore
X n 1 X
Z
1
1
B n