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Unformatted text preview: 202 Chapter 6 Sampling Distributions 3. When the underlying distribution is normal with mean ,u and variance 0 2 ’ calculate exact probabilities for X using the normal distribution with mean 0.2 pt and variance —— n 4. Apply the central limit theorem, wlgn the sample size is large, to approx
imate the sampling distribution of X by a normal distribution with mean 02 u. and variance —. n b
the standard normal probability P (Z <  . Don’t confuse the population distribution, which describes the variation for P(X§b)=P<Z§ b—u)
6J5 — 03%) Don’ts a single random variable, with the sampling distribution of a statistic. 2. When sampling from a _ﬁnite population of size N, don’t use U/ﬁ as the ;
standard deviation of X unless the ﬁnite population correction factor is i nearly 1. 3. When the population distribution is noticgbly nonnormal, don’t try to'
conclude that the sampling distribution of X is normal unless the sample size is at least moderately large, 30 or more. Review Exercises 6.51 6.52 6.53 6.54 The panel for a national science fair wishes to select
10 states from which a student representative will be
chosen at random from the students participating in the
state science fair. (a) Use Table 7 to select the 10 states. (b) Does the total selection process give each student
who participates in some state science fair an equal
chance of being selected to be a representative at
the national science fair? How many different samples of size n = 2 can be cho
sen from a ﬁnite population of size (a) N = 7; (b) N = 18? With reference to Exercise 6.52, what is the probability of choosing each sample in part (a) and the probability
of choosing each sample in part (b), if the samples are
to be random? Referring to Exercise 6.52, ﬁnd the value of the ﬁnite
population correction factor in the formula for 0% for part (a) and part (b). 6.55 The time to check out and process payment info m
tion at an ofﬁce supplies Web site can be modeled a:
random variable with mean [L = 63 seconds and a
ance 02 = 81. If the sample mean Y will be has w
a random sample of n = 36 times, what can we .
about the probability of getting a sample mean :4 .,
than 66.75, if we use (a) Chebyshev’s theorem;
(b) the central limit theorem? 6.56 The number of pieces of mail that a departrnen
ceives each day can be modeled by a distribution 1}
ing mean 44 and standard deviation 8. For a m”
sample of 35 days, what can be said about the prob If;
ity that the sample mean will be less than 40 or y than 48 using
(a) Chebyshev’s theorem;
(b) the central limit theorem? 6.57 If measurements of the speciﬁc gravity of a me : E
be looked upon as a sample from a normal w; tion having a standard deviation of 0.04, what '3 i The probability P( X 5 b) is approximately equal to 3 probability that the mean of a random sample of size
25 will be “off” by at most 0.02? ‘8 Adding graphite to iron can improve its ductile qual ities. If measurements of the diameter of graphite
spheres within an iron matrix can be modeled as a nor
mal distribution having standard deviation 0.16, what
is the probability that the mean of a sample of size
36 will differ from the population mean by more than 0.06? ".59 If 2 independent random samples of size 111 = 9 and 6.62 M = 16 are taken from a normal population, what is
the probability that the variance of the ﬁrst sample will ~ be at least 4 times as large as the variance of the second sample? If 2 independent sample of sizes 111 = 26 and 112 = 8
are taken from a normal population, what is the prob
ability that the variance of the second sample will be
at least 2.4 times the variance of the ﬁrst sample? When we sample from an inﬁnite population, what
happens to the standard error of the mean if the sample
size is (a) increased from 100 to 200; (b) increased from 200 to 300; (c) decreased from 360 to 90? A trafﬁc engineer collects data on trafﬁc ﬂow at a
busy intersection during the rush hour by recording the
number of westbound cars that are waiting for a green
light. The observations are made for each light change. 6.63 6.64 Key Terms 203 Explain why this sampling technique will not lead to
a random sample. Explain why the following may not lead to random
samples from the desired populations: (a) To determine the smoothness of shafts, a manu
facturer measures the roughness of the ﬁrst piece
made each morning. (b) To determine the mix of cars, trucks, and buses
in the rush hour, an engineer records the type of
vehicle passing a ﬁxed point at 1minute intervals. Several pickers are each asked to gather 30 ripe apples
and put them in a bag. (a) Would you expect all of the bags to weigh the
same? For one bag, let X1 be the weight of the
ﬁrst apple, X 2 the weight of the second apple, and
so on. Relate the weight of this bag, 30
2x:
i=1 to the approximate sampling distribution of Y. (b) Explain how your answer to part (a) leads to
the sampling distribution for the variation in bag weights.
(C) If the weight of an individual apple has mean ,u =
02 pound and standard deviation 0 = 0.03 pound,
ﬁnd the probability that the total weight of the bag
will exceed 6.2 pounds. Central limit theorem 184 Inﬁnite population 175 Chi square distributlon 189 Law of large numbers 182
Convolution formula 199 Moment generating function
Degrees of freedom 188 method 195 Discrete uniform distribution 179 Parameter 178 Distribution function method 197
: F distribution 190 Population 175 ' Finite population 175
Finite population correction Random sample
factor 1 8 1 Representation 192 Probability integral transformation 197
Probable error of the mean
176 Sample 175 Sampling distribution 180
Standard error of the mean 183
Standardized sample mean 183 Statistic 178
tdistribution 187 Transformation method 198
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 Fall '08
 Mendell

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