Page 304 - Applied statistics and probability for engineers

282 Chapter 8/Statistical intervals for a single sample

8-15. A civil engineer is analyzing the compressive 15.2 14.2 14.0 12.2 14.4 12.5
strength of concrete. Compressive strength is normally distrib- 14.3 14.2 13.5 11.8 15.2
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uted with σ = 1000(psi) . A random sample of 12 specimens
has a mean compressive strength of x = 3250 psi. Assume that the standard deviation is known to be σ = 0.5.
(a) Construct a 95% two-sided conidence interval on mean (a) Construct a 99% two-sided conidence interval on the mean
compressive strength. temperature.
(b) Construct a 99% two-sided conidence interval on mean (b) Construct a 95% lower-conidence bound on the mean
compressive strength. Compare the width of this coni- temperature.
dence interval with the width of the one found in part (a). (c) Suppose that you wanted to be 95% conident that the error
8-16. Suppose that in Exercise 8-14 we wanted the error in in estimating the mean temperature is less than 2 degrees
estimating the mean life from the two-sided conidence interval to Celsius. What sample size should be used?
be ive hours at 95% conidence. What sample size should be used? (d) Suppose that you wanted the total width of the two-sided
8-17. Suppose that in Exercise 8-14 you wanted the total conidence interval on mean temperature to be 1.5 degrees
width of the two-sided conidence interval on mean life to be Celsius at 95% conidence. What sample size should be used?
six hours at 95% conidence. What sample size should be used? 8-22. Ishikawa et al. (Journal of Bioscience and Bioengineering,
8-18. Suppose that in Exercise 8-15 it is desired to esti- 2012) studied the adhesion of various bioilms to solid surfaces
mate the compressive strength with an error that is less than 15 for possible use in environmental technologies. Adhesion assay
psi at 99% conidence. What sample size is required? is conducted by measuring absorbance at A . Suppose that for
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8-19. By how much must the sample size n be increased if the bacterial strain Acinetobacter, ive measurements gave read-
the length of the CI on μ in Equation 8-5 is to be halved? ings of 2.69, 5.76, 2.67, 1.62 and 4.12 dyne-cm . Assume that the
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8-20. If the sample size n is doubled, by how much is the standard deviation is known to be 0.66 dyne-cm .
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length of the CI on μ in Equation 8-5 reduced? What happens (a) Find a 95% conidence interval for the mean adhesion.
to the length of the interval if the sample size is increased by a (b) If the scientists want the conidence interval to be no
factor of four? wider than 0.55 dyne-cm , how many observations should
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8-21. An article in the Journal of Agricul- they take?
tural Science [“The Use of Residual Maximum Likelihood to 8-23. Dairy cows at large commercial farms often receive
Model Grain Quality Characteristics of Wheat with Variety, injections of bST (Bovine Somatotropin), a hormone used to
Climatic and Nitrogen Fertilizer Effects” (1997, Vol. 128, pp. spur milk production. Bauman et al. (Journal of Dairy Science,
135–142)] investigated means of wheat grain crude protein 1989) reported that 12 cows given bST produced an average of
content (CP) and Hagberg falling number (HFN) surveyed in 28.0 kg/d of milk. Assume that the standard deviation of milk
the United Kingdom. The analysis used a variety of nitrogen production is 2.25 kg/d.
fertilizer applications (kg N/ha), temperature (ºC), and total
monthly rainfall (mm). The following data below describe (a) Find a 99% conidence interval for the true mean milk
temperatures for wheat grown at Harper Adams Agricultural production.
College between 1982 and 1993. The temperatures measured (b) If the farms want the conidence interval to be no wider than
in June were obtained as follows: ±1.25 kg/d, what level of conidence would they need to use?

8-2 Confidence Interval on the Mean of a Normal
Distribution, Variance Unknown

When we are constructing conidence intervals on the mean μ of a normal population when
σ is known, we can use the procedure in Section 8-1.1. This CI is also approximately valid
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(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 8-1.5, we can
even handle the case of unknown variance for the large-sample-size situation. However, when
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the sample is small and σ is unknown, we must make an assumption about the form of the
underlying 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 conidence interval procedures of wide applicability. In
fact, moderate departure from normality will have little effect on validity. When the assump-
tion is unreasonable, an alternative is to use nonparametric statistical procedures that are valid
for any underlying distribution.