8
Random
8.1
8.2
8.3
8.4
8.5
Sampling, Large
Introduction
Distribution of
Distribution of
Distribution of
Standard Error
Sample Theory
the Sample Mean
the Sample Standard Deviation
the Difference Between Two Sample Means
EXERCISES
1. Verify Formula (8-4) for N = 4 and n - 2.
2. Verify Formula (8-4) for N = 4 and « = 3.
3. Verify Formula (8-4) for N = 5 and n = 4.
4. Verify Formula (8-4) for N = 5 and n = 3. \
5. For the case where N= 3 and n = 2, show that
3
 ( X i  m)² 
t 1

2 3 2
X t  (X 1 X 2  X 1 X 3  X 2 X 3  .


3  t 1

6. For the case where N = 3 and N = 2, show that
3
(X
i 1
i
 m)² 

1 3 2
X i  ( X1 X 2  X1 X 3  X 2 X 3  .


6  i1

7. Verify Formula (8-5) for TV = 4 and n = 2.
8. Verify Formula (8-5) for N = 4 and « = 3.
9. In a normal population with mean 72.10 and standard deviation 3.10, find the
probability that in a sample of 90 variates the mean will be less than 71.70.
10. In a savings bank the average account is $159.32 with a standard deviation
of $18.00. What is the probability that a group of 400 accounts taken at random
shows an average deposit of $160.00 or more?
11. In a particular area the daily wages of coal miners are normally distributed
with a mean of $16.50 and a standard deviation of $1.50. What is the
probability that a representative sample of 25 miners will have an average daily
wage below $15.75?
12. The heights of a certain group of adults have a mean of 67.42 inches and a
standard deviation of 2.58 inches. If these heights are normally distributed,
and if 25 people are taken at random from the group, what is the probability
that their mean will be 68.00 inches or more?
13. A random sample 100 variates is drawn from a normal population with mean 50
and standard deviation 8, and a random sample of 400 variates is drawn from a
normal population with mean 40 and standard deviation 12. Find the probability
that
(a) the mean of the first sample exceeds the mean of the second sample by 8 or
more;
(b) the two means differ in absolute value by 12 or more.
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14. From each of two normal and independent populations with identical means and
with standard deviations of 6.40 and 7.20, a sample of 64 variates is drawn.
Find the probability that the difference between the means of the samples
exceeds 0.60 in absolute value.
15. If a random sample of 36 variates is taken from a population, how many
variates should be taken in another sample from the same population to make the
standard error of the mean of the second sample 2/3 of the standard error of the
mean of the first sample?
16. If two random samples are taken from a population, and if the standard error
of the mean of one of them is k times the standard error of the mean of the
other, how are the sizes of the samples related?
17. In one restaurant the average amount spent for breakfast is 98.3 cents with
a standard deviation of 15.0 cents, and in a second restaurant the corresponding
figures are 92.4 and 12.0 cents. If a random sample of 80 breakfast charges is
taken from the first restaurant and a random sample of 60 from the second,
what is the probability that the difference between the average amounts of these
two samples is less than 1 cent in absolute value?
18. It is known that in a certain large city persons eating in a restaurant
spend, on the average, 98.3 cents for breakfast with a standard deviation of
15.0 cents. If each of 50 restaurants is asked to select at random the
breakfast bills of 100 persons and to report the average amount spent by these
100 persons, how many restaurants may be expected to report the average bill as
$1.00 or more?
19. The average gasoline mileage for cars of make A is 20 miles per gallon with
a standard deviation of 6 miles per gallon. Comparable figures for cars of make
B are 25 and 5.5 miles per gallon. Assume that the gasoline mileage for each of
the two makes is normally distributed. What is the probability in a gasoline
mileage contest that the average gasoline mileage for 10 cars of make A is
greater than that for 9 cars of make B?
20. On the average, students from university A get up 50 minutes after sunrise
with a standard deviation of 15 minutes, students from university B get up 60
minutes after sunrise with a standard deviation of 18 minutes. A group of 25
students from university A takes a trip with a group of 20 students from
university B. Find the probability that the average time of rising for the group
from university B is earlier than that for the group from university A.
21. Past experience shows that, on the average, men and women students perform
equally well in tests on statistics. The standard deviation of the men students
however, is found to be 15; that of the women students 12. What is the
probability that, in an examination in statistics taken by 120 students of which
69 are men, the average of the women students will exceed that of the men by
more than 3.0 points?
22. Two normal populations with the same variance σ² = 4 have means of 66.0 and
65.5. What is the probability that the mean of a sample of 50 variates from the
first population exceeds the mean of a sample of 50 variates from the second
population?
23. If all possible samples of size 25 are drawn from a normally distributed
population with mean equal to 20 and standard deviation equal to 4, within what
range will the middle 90% of the sample means fall?
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24. The daily wages in a particular industry are normally distributed with a
mean of $13.20. If 9% of the mean daily wages of samples of 25 workers fall
below $12.53, what is the standard deviation of daily wages in this industry?
25. From a normal distribution all possible samples of size 10 are selected.
If 2% of these samples have means which differ from the mean of the population
by more than 4.00 in absolute value, find the standard deviation of the
population.
26. In a normal distribution of a continuous variable with mean 9.00 the
probability that the mean of a random sample of 25 variates exceeds 10.00 is
0.33. What is the probability that a single variate selected at random from
this distribution is greater than 10.00?
27. In a normal population with mean 8.00, the probability that the mean of a
random sample of 25 variates exceeds 9.76 is 0.33. What is the probability that
a single variate selected at random from the population is greater than 3.00?
28. If it is assumed that the heights of men are normally distributed with a
standard deviation of 2.5 inches, how large a sample should be taken in order to
be fairly sure (probability 0.95) that the sample mean does not differ from the
true mean (population mean) by more than 0.50 in absolute value?
29. Referring to the data of Exercise 28, how large a sample should be taken if
we want to be 99% sure that the sample mean does not differ from the true mean
by more than 0.50 in absolute value?
30. A normal population has a standard deviation of 2.0. How large a sample
should be taken in order to be 99% sure that the sample mean differs from the
population mean by less than 0.8?
31. In measuring reaction time, a psychologist estimates that the standard
deviation of the population is 0.05 seconds. How large a sample of measurements
must he take in order to be 99% sure that he will estimate the mean of the
population with an error not exceeding 0.01 seconds?
32. IQ scores are usually accepted to be normally distributed with a standard
deviation of 15. At least how large a sample should be taken in order to be 95%
sure that the sample mean does not differ from the population mean by more than
2 IQ points?
33. On a nation-wide examination the scores followed a normal distribution with
a mean of 72 and a standard deviation of 10. How large a sample of candidates
from University Y must be taken in order that there be a 90 percent chance that
its mean score is more than 70%?
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