Page 312 - Applied statistics and probability for engineers

290 Chapter 8/Statistical intervals for a single sample

or
19 .
σ ≤ ( )0 0153 = 0 0287 (fluid ounce ) 2
2
.
10 117
.

This last expression may be converted into a conidence interval on the standard deviation σ by taking the square root
of both sides, resulting in
σ ≤ 0.17
Practical Interpretation: Therefore, at the 95% level of conidence, the data indicate that the process standard devia-

tion could be as large as 0.17 luid ounce. The process engineer or manager now needs to determine whether a standard

deviation this large could lead to an operational problem with under- or over-i lled bottles.

Exercises FOR SECTION 8-3

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion

8-46. Determine the values of the following percentiles: 8-53. An article in Urban Ecosystems, “Urbani-
2
2
2
2
2
2
χ .0 05 10, , χ .0 025 15, , χ .0 0112, , χ .0 95 20, , χ .0 99 18, , χ .0 995,116 , and χ . 2 0 005 25 . zation and Warming of Phoenix (Arizona, USA): Impacts, Feed-
,
backs and Mitigation” (2002, Vol. 6, pp. 183–203), mentions
2
8-47. Determine the χ percentile that is required to construct that Phoenix is ideal to study the effects of an urban heat island
each of the following CIs: because it has grown from a population of 300,000 to approxi-
(a) Coni dence level = 95%, degrees of freedom = 24, one- mately 3 million over the last 50 years, which is a period with a
sided (upper) continuous, detailed climate record. The 50-year averages of the

(b) Conidence level = 99%, degrees of freedom = 9, one-sided mean annual temperatures at eight sites in Phoenix follow. Check
(lower) the assumption of normality in the population with a probability

(c) Conidence level = 90%, degrees of freedom = 19, two-sided.

plot. Construct a 95% conidence interval for the standard devia-
8-48. A rivet is to be inserted into a hole. A random sample tion over the sites of the mean annual temperatures.
of n = 15 parts is selected, and the hole diameter is measured.
Average Mean
The sample standard deviation of the hole diameter measure- Site Temperature (°C)
ments is s = 0.008 millimeters. Construct a 99% lower coni -
2
dence bound for σ . Sky Harbor Airport 23.3
8-49. Consider the situation in Exercise 8-48. Find a 99% Phoenix Greenway 21.7

lower conidence bound on the standard deviation. Phoenix Encanto 21.6
8-50. The sugar content of the syrup in canned peaches is Waddell 21.7
normally distributed. A random sample of n = 10 cans yields
a sample standard deviation of s = 4.8 milligrams. Calculate a Litchi eld 21.3
95% two-sided conidence interval for σ. Laveen 20.7

8-51. The percentage of titanium in an alloy used in aero- Maricopa 20.9
space castings is measured in 51 randomly selected parts. The Harlquahala 20.1
sample standard deviation is s = 0.37. Construct a 95% two-

sided conidence interval for σ. 8-54. An article in Cancer Research [“Analyses of Lit-
8-52. An article in Medicine and Science in Sports and ter-Matched Time-to-Response Data, with Modii cations for
Exercise [“Electrostimulation Training Effects on the Physi- Recovery of Interlitter Information” (1977, Vol. 37, pp. 3863–
cal Performance of Ice Hockey Players” (2005, Vol. 37, pp. 3868)] tested the tumorigenesis of a drug. Rats were randomly
455–460)] considered the use of electromyostimulation (EMS) selected from litters and given the drug. The times of tumor
as a method to train healthy skeletal muscle. EMS sessions appearance were recorded as follows:
consisted of 30 contractions (4-second duration, 85 Hz) and 101, 104, 104, 77, 89, 88, 104, 96, 82, 70, 89, 91, 39, 103, 93,
were carried out three times per week for three weeks on 17 ice 85, 104, 104, 81, 67, 104, 104, 104, 87, 104, 89, 78, 104, 86,
hockey players. The 10-meter skating performance test showed 76, 103, 102, 80, 45, 94, 104, 104, 76, 80, 72, 73
a standard deviation of 0.09 seconds. Construct a 95% coni -

dence interval of the standard deviation of the skating perfor- Calculate a 95% conidence interval on the standard deviation
mance test. of time until a tumor appearance. Check the assumption of