Answered: 1 A random variable, x, has a standard…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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A random variable, x, has a standard normal distribution. Calculate the probability that x lies in the following intervals: (a) (0.25,0.75) ® (~03.0.1) (c) within 1.5 standard deviations of the mean (d) more than two standard deviations from the mean (@ (~1.7.-02) A random variable, x, has a normal distribution with ‘mean 4 and standard deviation 0.8. Calculate the probability that @ 3046 @ x0 © 16 The scores from IQ tests have a mean of 100 and a standard deviation of 15. What should a person score in order to be described as in the top 10% of the population? A machine produces car pistons. The diameter of the pistons follows a normal distribution, mean 6.04 cm with a standard deviation of 0.02 cm. The piston is acceptable if its diameter is in the range 6.010 cm to 6.055 cm. What percentage of pistons is acceptable? 1€ The random variable, x. has a normal distribution. How many standard deviations above the mean must the point P be placed if the tail-end is to represent @ 10% (b) 5% () 1% of the total area? (See Figure 29.20.) Nx) Figure 29.20 Graph for Question 6. Consider Figure 29.21. The two tail-ends have equal area. How many standard deviations from the mean must A and B be placed if the tail-ends are (a) 10% (b) 5% () 1% of the total area? Nx) Figure 29.21 Graph for Question 7. . (a) 0.1747 (b) 0.1577 (c) 0.8664 (d) 0.0455 (e) 03762 . (a) 0.5858 (b) 0.4199 (c) 0.2266 (d) 0.5987 (e) 0.5467 3 @ 04436 () 06554 () 02604 (d) 05544 & 1 & n% & @128 ® 1685 7 @ies ®196 (©233 (©2.57

EXERCISES 29.14
A random variable, x, has a standard normal
distribution. Calculate the probability that x lies in the
following intervals:
The random variable, x, has a normal distribution.
How many standard deviations above the mean must
the point P be placed if the tail-end is to represent
(a) (0.25, 0.75)
(a) 10%
(b) 5%
(c) 1%
(b) (-0.3, 0.1)
of the total area? (See Figure 29.20.)
(c) within 1.5 standard deviations of the mean
(d) more than two standard deviations from the mean
N(x) A
(e) (-1.7, -0.2)
A random variable, x, has a normal distribution with
mean 4 and standard deviation 0.8. Calculate the
probability that
(a) 3.0 <x< 4.4
(b) 2.5 <x < 3.9
Figure 29.20
Graph for Question 6.
(c) x> 4.6
(d) x< 4.2
(e) x is within 0.6 of the mean
Consider Figure 29.21. The two tail-ends have equal
area. How many standard deviations from the
must A and B be placed if the tail-ends are
A random variable, t, has a normal distribution with
ean
mean 1 and standard deviation 2.5. Calculate the
probability that
(a) -1 <1<2
(a) 10%
(b) 5%
(c) 1%
(b) t >0
of the total area?
(c) t| < 0.9
(d) t| > 1.6
The scores from IQ tests have a mean of 100 and a
standard deviation of 15. What should a person score
in order to be described as in the top 10% of the
population?
N(x) A
5 A machine produces car pistons. The diameter of the
pistons follows a normal distribution, mean 6.04 cm
with a standard deviation of 0.02 cm. The piston is
acceptable if its diameter is in the range 6.010 cm to
6.055 cm. What percentage of pistons is
ассeptable?
A
B
Figure 29.21
Graph for Question 7.
Solutions
(a) 0.1747
(d) 0.0455
(b) 0.1577
(c) 0.8664
119
(e) 0.3762
71%
2 (a) 0.5858
(d) 0.5987
(b) 0.4199
(c) 0.2266
(e) 0.5467
6 (a) 1.28
(b) 1.645
(c) 2.33
3 (a) 0.4436
(b) 0.6554
(c) 0.2604
7 (a) 1.64
(b) 1.96
(c) 2.57
(d) 0.5544

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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1 A random variable, x, has a standard normal distribution. Calculate the probability that x lies in the following intervals: (a) (0.25, 0.75) (b) (-0.3, 0.1) (c) within 1.5 standard deviations of the mean (d) more than two standard deviations from the mean (e) (-1.7, –0.2)