Page 270 - Applied statistics and probability for engineers

248 Chapter 7/Point Estimation of Parameters and Sampling Distributions

Exercises FOR SECTION 7-2

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
7-1. Consider the hospital emergency room data from Exercise (a) Less than 10 minutes (b) Between 5 and 10 minutes
6-124. Estimate the proportion of patients who arrive at this (c) Less than 6 minutes
emergency department experiencing chest pain. 7-13. A random sample of size n 1 16= is selected from
7-2. Consider the compressive strength data in Table 6-2. a normal population with a mean of 75 and a standard devia-
What proportion of the specimens exhibit compressive strength tion of 8. A second random sample of size n 2 = 9 is taken from
of at least 200 psi? another normal population with mean 70 and standard devia-
7-3. PVC pipe is manufactured with a mean diameter of 1.01 tion 12. Let X 1 and X 2 be the two sample means. Find:
inch and a standard deviation of 0.003 inch. Find the probability (a) The probability that X 1 − X 2 exceeds 4
.
that a random sample of n = 9 sections of pipe will have a sample (b) The probability that 3 5. ≤ X 1 − X 2 ≤ 5 5
mean diameter greater than 1.009 inch and less than 1.012 inch. 7-14. A consumer electronics company is comparing the bright-
7-4. Suppose that samples of size n = 25 are selected at ness of two different types of picture tubes for use in its television
random from a normal population with mean 100 and standard sets. Tube type A has mean brightness of 100 and standard devia-
deviation 10. What is the probability that the sample mean falls tion of 16, and tube type B has unknown mean brightness, but the
1
in the interval from μ − . σ1 8 X to μ + . σ ? standard deviation is assumed to be identical to that for type A
0
X
X
X
7-5. A synthetic iber used in manufacturing carpet has . A random sample of n = 25 tubes of each type is selected, and
tensile strength that is normally distributed with mean 75.5 psi X B − X A is computed. If μ B equals or exceeds μ A , the manufac-
and standard deviation 3.5 psi. Find the probability that a ran- turer would like to adopt type B for use. The observed difference
dom sample of n = 6 iber specimens will have sample mean is x B − x = 3 5. . What decision would you make, and why?
A
tensile strength that exceeds 75.75 psi. 7-15. The elasticity of a polymer is affected by the con-
7-6. Consider the synthetic iber in the previous exercise. centration of a reactant. When low concentration is used, the
How is the standard deviation of the sample mean changed true mean elasticity is 55, and when high concentration is used,
when the sample size is increased from n = 6 to n = 49? the mean elasticity is 60. The standard deviation of elasticity is
7-7. The compressive strength of concrete is normally dis- 4 regardless of concentration. If two random samples of size 16
tributed with μ = 2500 psi and σ = 50 psi. Find the probability are taken, ind the probability that X high − X low • 2 .
that a random sample of n = 5 specimens will have a sample 7-16. Scientists at the Hopkins Memorial Forest in western
mean diameter that falls in the interval from 2499 psi to 2510 psi. Massachusetts have been collecting meteorological and environ-
7-8. Consider the concrete specimens in Exercise 7-7. mental data in the forest data for more than 100 years. In the past
What is the standard error of the sample mean? few years, sulfate content in water samples from Birch Brook
7-9. A normal population has mean 100 and variance 25. has averaged 7.48 mg/L with a standard deviation of 1.60 mg/L.
How large must the random sample be if you want the standard (a) What is the standard error of the sulfate in a collection of
error of the sample average to be 1.5? 10 water samples?
7-10. Suppose that the random variable X has the continu- (b) If 10 students measure the sulfate in their samples, what is
ous uniform distribution the probability that their average sulfate will be between
⎧1 , 0 ≤ x ≤1 6.49 and 8.47 mg/L?
f x ( ) = ⎨ (c) What do you need to assume for the probability calculated
⎩ , 0 otherwise
in (b) to be accurate?
Suppose that a random sample of n = 12 observations is selected 7-17. From the data in Exercise 6-21 on the pH of rain in
from this distribution. What is the approximate probability dis- Ingham County, Michigan:
tribution of X − 6 ? Find the mean and variance of this quantity.
7-11. Suppose that X has a discrete uniform distribution 5.47 5.37 5.38 4.63 5.37 3.74 3.71 4.96 4.64 5.11 5.65
5.39 4.16 5.62 4.57 4.64 5.48 4.57 4.57 4.51 4.86 4.56
⎧ 1 , x = 1 2
⎪
, ,3
f x ( ) = ⎨ 3 4.61 4.32 3.98 5.70 4.15 3.98 5.65 3.10 5.04 4.62 4.51
⎩ ⎪ , 0 otherwise 4.34 4.16 4.64 5.12 3.71 4.64
A random sample of n = 36 is selected from this population. What proportion of the samples has pH below 5.0?
Find the probability that the sample mean is greater than 2.1 7-18. Researchers in the Hopkins Forest (see Exercise 7-16)
but less than 2.5, assuming that the sample mean would be also count the number of maple trees (genus acer) in plots
measured to the nearest tenth. throughout the forest. The following is a histogram of the
7-12. The amount of time that a customer spends waiting number of live maples in 1002 plots sampled over the past 20
at an airport check-in counter is a random variable with mean years. The average number of maples per plot was 19.86 trees
8.2 minutes and standard deviation 1.5 minutes. Suppose that a with a standard deviation of 23.65 trees.
random sample of n = 49 customers is observed. Find the prob- (a) If we took the mean of a sample of eight plots, what would
ability that the average time waiting in line for these customers is be the standard error of the mean?