Page 308 - Applied statistics and probability for engineers

286 Chapter 8/Statistical intervals for a single sample

8-26. Determine the t-percentile that is required to con- subjected to loads, and the resulting frequencies (in hertz) were
struct each of the following one-sided conidence intervals: as follows:
(a) Conidence level = 95%, degrees of freedom = 14 230.66, 233.05, 232.58, 229.48, 232.58
(b) Conidence level = 99%, degrees of freedom = 19 Check the assumption of normality in the population. Calculate
(c) Conidence level = 99.9%, degrees of freedom = 24 a 90% two-sided conidence interval on mean natural frequency.
8-27. A random sample has been taken from a normal distri- 8-35. The Bureau of Meteorology of the Australian Gov-
bution. Output from a software package follows: ernment provided the mean annual rainfall (in millimeters)
in Australia 1983–2002 as follows (http://www.bom.gov.au/
Variable N Mean SE Mean StDev Varianc Sum
climate/change/rain03.txt):
x 10 ? 0.507 1.605 ? 251.848
499.2, 555.2, 398.8, 391.9, 453.4, 459.8, 483.7, 417.6, 469.2,
(a) Fill in the missing quantities. 452.4, 499.3, 340.6, 522.8, 469.9, 527.2, 565.5, 584.1, 727.3,
(b) Find a 95% CI on the population mean. 558.6, 338.6
8-28. A random sample has been taken from a normal distri- Check the assumption of normality in the population. Con-
bution. Output from a software package follows: struct a 95% conidence interval for the mean annual rainfall.
Variable N Mean SE Mean StDev Varianc Sum 8-36. The solar energy consumed (in trillion
BTU) in the United States by year from 1989 to 2004 (source:
x ? ? 1.58 6.11 ? 751.40
U.S. Department of Energy, http://www.eia.doe.gov/emeu) is
(a) Fill in the missing quantities. shown in the following table. Read down then across for year.
(b) Find a 95% CI on the population mean.
55.291 59.718 62.688 63.886
8-29. A research engineer for a tire manufacturer is investigat- 66.458 68.548 69.857 70.833
ing tire life for a new rubber compound and has built 16 tires and 70.237 69.787 68.793 66.388
tested them to end-of-life in a road test. The sample mean and 65.454 64.391 63.62 63.287
standard deviation are 60,139.7 and 3645.94 kilometers. Find a Check the assumption of normality in the population. Construct
95% conidence interval on mean tire life. a 95% conidence interval for the mean solar energy consumed.
8-30. An Izod impact test was performed on 20 specimens 8-37. The brightness of a television picture tube can be evalu-
of PVC pipe. The sample mean is x = 1.25 and the sample ated by measuring the amount of current required to achieve
standard deviation is s = 0.25. Find a 99% lower conidence a particular brightness level. A sample of 10 tubes results in
bound on Izod impact strength. x = 317 .2 and s = 15.7. Find (in microamps) a 99% conidence
8-31. A postmix beverage machine is adjusted to release interval on mean current required. State any necessary assump-
a certain amount of syrup into a chamber where it is mixed tions about the underlying distribution of the data.
with carbonated water. A random sample of 25 beverages was 8-38. A particular brand of diet margarine was analyzed to
found to have a mean syrup content of x = 1 10. luid ounce and determine the level of polyunsaturated fatty acid (in percent-
a standard deviation of s = 0.015 luid ounce. Find a 95% CI on ages). A sample of six packages resulted in the following data:
the mean volume of syrup dispensed. 16.8, 17.2, 17.4, 16.9, 16.5, 17.1.
8-32. An article in Medicine and Science in Sports and Exer- (a) Check the assumption that the level of polyunsaturated
cise [“Maximal Leg-Strength Training Improves Cycling Econ- fatty acid is normally distributed.
omy in Previously Untrained Men” (2005, Vol. 37, pp. 131–136)] (b) Calculate a 99% conidence interval on the mean μ. Provide
studied cycling performance before and after eight weeks of leg- a practical interpretation of this interval.
strength training. Seven previously untrained males performed (c) Calculate a 99% lower conidence bound on the mean.
leg-strength training three days per week for eight weeks (with Compare this bound with the lower bound of the two-sided
four sets of ive replications at 85% of one repetition maximum). conidence interval and discuss why they are different.
Peak power during incremental cycling increased to a mean of 315 8-39. The compressive strength of concrete is being tested by
watts with a standard deviation of 16 watts. Construct a 95% con- a civil engineer who tests 12 specimens and obtains the fol-
idence interval for the mean peak power after training. lowing data:
8-33. An article in Obesity Research [“Impaired Pressure
Natriuresis in Obese Youths” (2003, Vol. 11, pp. 745–751)] 2216 2237 2249 2204
described a study in which all meals were provided for 14 lean 2225 2301 2281 2263
boys for three days followed by one stress test (with a video- 2318 2255 2275 2295
game task). The average systolic blood pressure (SBP) during (a) Check the assumption that compressive strength is normally
the test was 118.3 mm HG with a standard deviation of 9.9 mm distributed. Include a graphical display in your answer.
HG. Construct a 99% one-sided upper conidence interval for (b) Construct a 95% two-sided conidence interval on the mean
mean SBP. strength.
8-34. An article in the Journal of Composite Materials (c) Construct a 95% lower conidence bound on the mean
(December 1989, Vol. 23, p. 1200) describes the effect of strength. Compare this bound with the lower bound of the
delamination on the natural frequency of beams made from two-sided conidence interval and discuss why they are
composite laminates. Five such delaminated beams were different.