8-3 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN 2578-10. The diameter of holes for cable harness is known to
have a normal distribution with 0.01 inch. A random
sample of size 10 yields an average diameter of 1.5045 inch.
Find a 99% two-sided confidence interval on the mean hole
diameter.
8-11. A manufacturer produces piston rings for an auto-
mobile engine. It is known that ring diameter is normally dis-
tributed with 0.001 millimeters. A random sample of 15
rings has a mean diameter of millimeters.
(a) Construct a 99% two-sided confidence interval on the
mean piston ring diameter.
(b) Construct a 95% lower-confidence bound on the mean
piston ring diameter.
8-12. The life in hours of a 75-watt light bulb is known to be
normally distributed with 25 hours. A random sample of
20 bulbs has a mean life of hours.
(a) Construct a 95% two-sided confidence interval on the
mean life.
(b) Construct a 95% lower-confidence bound on the mean
life.
8-13. A civil engineer is analyzing the compressive strength
of concrete. Compressive strength is normally distributed with
^2 1000(psi)^2. A random sample of 12 specimens has a
mean compressive strength of x 3250 psi.x 1014x74.036(a) Construct a 95% two-sided confidence interval on mean
compressive strength.
(b) Construct a 99% two-sided confidence interval on mean
compressive strength. Compare the width of this confi-
dence interval with the width of the one found in part (a).
8-14. Suppose that in Exercise 8-12 we wanted to be 95%
confident that the error in estimating the mean life is less than
five hours. What sample size should be used?
8-15. Suppose that in Exercise 8-12 we wanted the total
width of the two-sided confidence interval on mean life to be
six hours at 95% confidence. What sample size should be
used?
8-16. Suppose that in Exercise 8-13 it is desired to estimate
the compressive strength with an error that is less than 15 psi
at 99% confidence. What sample size is required?
8-17. By how much must the sample size nbe increased if
the length of the CI on in Equation 8-7 is to be halved?
8-18. If the sample size nis doubled, by how much is the
length of the CI on in Equation 8-7 reduced? What happens
to the length of the interval if the sample size is increased by a
factor of four?8-3 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL
DISTRIBUTION, VARIANCE UNKNOWNWhen 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 nis 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
the sample is small and ^2 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
and unknown variance ^2. Assume that a random sample of size n, say X 1 , X 2 , p, Xn, is avail-
able, and let and S^2 be the sample mean and variance, respectively.
We wish to construct a two-sided CI on . If the variance ^2 is known, we know that
has a standard normal distribution. When ^2 is unknown, a logical pro-
cedure is to replace with the sample standard deviation S. The random variable Znow be-
comes. A logical question is what effect does replacing by Shave on the
distribution of the random variable T? If nis large, the answer to this question is “very little,”
and we can proceed to use the confidence interval based on the normal distribution fromT 1 X 2 1 S 1 n 2
Z 1 X 2 1 1 n 2
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