Page 317 - Applied statistics and probability for engineers

Section 8-4/Large-Sample Conidence Interval for a Population Proportion 295

Exercises FOR SECTION 8-4

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

8-59. The fraction of defective integrated circuits produced subjected to an impact test, and some damage was observed on
in a photolithography process is being studied. A random sample 18 of these helmets.
of 300 circuits is tested, revealing 13 defectives. (a) Find a 95% two-sided conidence interval on the true pro-
(a) Calculate a 95% two-sided CI on the fraction of defective portion of helmets that would show damage from this test.
circuits produced by this particular tool. (b) Using the point estimate of p from the 50 helmets, how
(b) Calculate a 95% upper conidence bound on the fraction of many helmets must be tested to be 95% conident that the
defective circuits. error in estimating p is less than 0.02?
(c) How large must the sample be if we wish to be at least 95%
8-60. An article in Knee Surgery, Sports Traumatology,
conident that the error in estimating p is less than 0.02
Arthroscopy [“Arthroscopic Meniscal Repair with an Absorba-
regardless of the true value of p?
ble Screw: Results and Surgical Technique” (2005, Vol. 13, pp.
8-65. The Arizona Department of Transportation wishes to
273–279)] showed that only 25 out of 37 tears (67.6%) located
between 3 and 6 mm from the meniscus rim were healed. survey state residents to determine what proportion of the popu-
(a) Calculate a two-sided 95% conidence interval on the pro- lation would like to increase statewide highway speed limits
portion of such tears that will heal. from 65 mph to 75 mph. How many residents does the depart-
(b) Calculate a 95% lower conidence bound on the proportion ment need to survey if it wants to be at least 99% conident that
of such tears that will heal. the sample proportion is within 0.05 of the true proportion?
8-66. A study is to be conducted of the percentage of
8-61. The 2004 presidential election exit polls from the
homeowners who own at least two television sets. How large a
critical state of Ohio provided the following results. The exit
sample is required if we wish to be 99% conident that the error
polls had 2020 respondents, 768 of whom were college gradu-
in estimating this quantity is less than 0.017?
ates. Of the college graduates, 412 voted for George Bush.
8-67. The U.S. Postal Service (USPS) has used optical character
(a) Calculate a 95% conidence interval for the proportion of
recognition (OCR) since the mid-1960s. In 1983, USPS began
college graduates in Ohio who voted for George Bush.
deploying the technology to major post ofices throughout the
(b) Calculate a 95% lower conidence bound for the proportion
country (www.britannica.com). Suppose that in a random sample
of college graduates in Ohio who voted for George Bush.
of 500 handwritten zip code digits, 466 were read correctly.
8-62. Of 1000 randomly selected cases of lung cancer,
(a) Construct a 95% conidence interval for the true proportion
823 resulted in death within 10 years.
of correct digits that can be automatically read.
(a) Calculate a 95% two-sided conidence interval on the death
(b) What sample sample size is needed to reduce the margin of
rate from lung cancer.
error to 1%?
(b) Using the point estimate of p obtained from the preliminary
(c) How would the answer to part (b) change if you had to assume
sample, what sample size is needed to be 95% conident that
that the machine read only one-half of the digits correctly?
the error in estimating the true value of p is less than 0.03?
8-68. Information on a packet of seeds claims that 93% of
(c) How large must the sample be if you wish to be at least
95% conident that the error in estimating p is less than them will germinate. Of the 200 seeds that I planted, only 180
germinated.
0.03, regardless of the true value of p?
(a) Find a 95% conidence interval for the true proportion of
8-63. An article in the Journal of the American Statistical seeds that germinate based on this sample.
Association (1990, Vol. 85, pp. 972–985) measured the weight (b) Does this seem to provide evidence that the claim is wrong?
of 30 rats under experiment controls. Suppose that 12 were
8-69. Use the data from Exercise 8-59 to compute a two-sided
underweight rats.
Agresti-Coull CI on the proportion of defective circuits. Com-
(a) Calculate a 95% two-sided conidence interval on the true
pare and discuss the CI to the one computed in Exercise 8-59.
proportion of rats that would show underweight from the
8-70. Use the data from Exercise 8-60 to compute a two-sided
experiment.
Agresti-Coull CI on the proportion of tears that will heal. Com-
(b) Using the point estimate of p obtained from the preliminary
pare and discuss the CI to the one computed in Exercise 8-60.
sample, what sample size is needed to be 95% conident that
8-71. Use the data from Exercise 8-66 to compute a two-sided
the error in estimating the true value of p is less than 0.02?
Agresti-Coull CI on the proportion of correct digits that can
(c) How large must the sample be if you wish to be at least be automatically read. Compare and discuss the CI to the one
95% conident that the error in estimating p is less than
computed in Exercise 8-66.
0.02, regardless of the true value of p?
8-72. Use the data from Exercise 8-68 to compute a two-sided
8-64. A random sample of 50 suspension helmets used Agresti-Coull CI on the proportion of seeds that germinate. Com-
by motorcycle riders and automobile race-car drivers was pare and discuss the CI to the one computed in Exercise 8-68.