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8-3 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN 257

8-10. The diameter of holes for cable harness is known to (a) Construct a 95% two-sided confidence interval on mean
have a normal distribution with 0.01 inch. A random compressive strength.
sample of size 10 yields an average diameter of 1.5045 inch. (b) Construct a 99% two-sided confidence interval on mean
Find a 99% two-sided confidence interval on the mean hole compressive strength. Compare the width of this confi-
diameter. dence interval with the width of the one found in part (a).
8-11. A manufacturer produces piston rings for an auto- 8-14. Suppose that in Exercise 8-12 we wanted to be 95%
mobile engine. It is known that ring diameter is normally dis- confident that the error in estimating the mean life is less than
tributed with 0.001 millimeters. A random sample of 15 five hours. What sample size should be used?
rings has a mean diameter of x 74.036 millimeters. 8-15. Suppose that in Exercise 8-12 we wanted the total
(a) Construct a 99% two-sided confidence interval on the width of the two-sided confidence interval on mean life to be
mean piston ring diameter. six hours at 95% confidence. What sample size should be
(b) Construct a 95% lower-confidence bound on the mean used?
piston ring diameter. 8-16. Suppose that in Exercise 8-13 it is desired to estimate
8-12. The life in hours of a 75-watt light bulb is known to be the compressive strength with an error that is less than 15 psi
normally distributed with 25 hours. A random sample of at 99% confidence. What sample size is required?
20 bulbs has a mean life of x 1014 hours. 8-17. By how much must the sample size n be increased if
(a) Construct a 95% two-sided confidence interval on the the length of the CI on in Equation 8-7 is to be halved?
mean life. 8-18. If the sample size n is doubled, by how much is the
(b) Construct a 95% lower-confidence bound on the mean length of the CI on in Equation 8-7 reduced? What happens
life.
to the length of the interval if the sample size is increased by a
8-13. A civil engineer is analyzing the compressive strength factor of four?
of concrete. Compressive strength is normally distributed with
2
2
1000(psi) . A random sample of 12 specimens has a
mean compressive strength of x 3250 psi.

8-3 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL
DISTRIBUTION, VARIANCE UNKNOWN

When we are constructing confidence intervals on the mean of a normal population when
2
is known, we can use the procedure in Section 8-2.1. This CI is also approximately valid
(because of the central limit theorem) regardless of whether or not the underlying population
is normal, so long as n is reasonably large (n 40, say). As noted in Section 8-2.5, we can
even handle the case of unknown variance for the large-sample-size situation. However, when
2
the sample is small and is unknown, we must make an assumption about the form of the un-
derlying distribution to obtain a valid CI procedure. A reasonable assumption in many cases is
that the underlying distribution is normal.
Many populations encountered in practice are well approximated by the normal distribu-
tion, so this assumption will lead to confidence interval procedures of wide applicability. In
fact, moderate departure from normality will have little effect on validity. When the assump-
tion is unreasonable, an alternate is to use the nonparametric procedures in Chapter 15 that are
valid for any underlying distribution.
Suppose that the population of interest has a normal distribution with unknown mean
2
and unknown variance . Assume that a random sample of size n, say X , X , p , X , is avail-
n
2
1
2
able, and let X and S be the sample mean and variance, respectively.
2
We wish to construct a two-sided CI on . If the variance is known, we know that
2
Z 1X 2 1 1n2 has a standard normal distribution. When is unknown, a logical pro-
cedure is to replace with the sample standard deviation S. The random variable Z now be-
comes T 1X 2 1S 1n2 . A logical question is what effect does replacing by S have on the
distribution of the random variable T? If n is large, the answer to this question is “very little,”
and we can proceed to use the confidence interval based on the normal distribution from