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ssrs export to pdf barcode font Sampling Theory in Software
CHAPTER 5 Sampling Theory Reading QR Code 2d Barcode In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Generating QR Code In None Using Barcode encoder for Software Control to generate, create QR Code image in Software applications. They are not, however, mutually independent. Indeed, when j 2 k, the joint distribution of Xj and Xk is given by P(Xj al, Xk an) P(Xj 1 P(Xk N al)P(Xk an Z Xj an Z Xj al) l2n l n al) Decoding QR Code In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. QR Code Creation In C#.NET Using Barcode printer for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications. where l and n range from 1 to N.
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Barcode Decoder In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Print Barcode In Java Using Barcode encoder for BIRT reports Control to generate, create barcode image in BIRT applications. a a (al
l 1n 1
m)(an
m)P(Xj
al, Xk
1 1 a b a (a N N 1 l2n 1 l where the last sum contains a total of N(N and n. Now, by elementary algebra, c m)(an
1) terms, corresponding to all possible pairs of unequal l
[(a1
m)]2 a (al
a (al
l2n 1
m)(an
In this equation, the lefthand side is zero, since by definition a1 a2 c aN Nm
and the first sum on the righthand side equals, by definition, Ns2. Hence, a (al
l2n 1
m)(an
Ns2 s2 N 1
Cov (Xj, Xk) 1 1 a b( Ns2) N N 1
5.48. Prove that (a) the mean, (b) the variance of the sample mean in Problem 5.47 are given, respectively, by mX m c n
2 sX
s2 N n aN
n b 1
# E(X) X1 1 n (m
b m) 1 n [E(X1) m
E(Xn)] where we have used Problem 5.47(a). CHAPTER 5 Sampling Theory
(b) Using Theorems 35 and 316 (generalized), and Problem 5.47, we obtain # Var (X) 1 Var a a Xj b n2 j 1 1 c ns2 n2 s2 n c1 n N n(n 1 d 1 1 c Var (Xj) n2 ja 1 1)a s2 N 1 bd n b 1
a Cov (Xj, Xk) d
j2k 1
s2 N n aN
SUPPLEMENTARY PROBLEMS
Sampling distribution of means
5.49. A population consists of the four numbers 3, 7, 11, 15. Consider all possible samples of size two that can be drawn with replacement from this population. Find (a) the population mean, (b) the population standard deviation, (c) the mean of the sampling distribution of means, (d) the standard deviation of the sampling distribution of means. Verify (c) and (d) directly from (a) and (b) by use of suitable formulas. 5.50. Solve Problem 5.49 if sampling is without replacement. 5.51. The weights of 1500 ball bearings are normally distributed with a mean of 22.40 oz and a standard deviation of 0.048 oz. If 300 random samples of size 36 are drawn from this population, determine the expected mean and standard deviation of the sampling distribution of means if sampling is done (a) with replacement, (b) without replacement. 5.52. Solve Problem 5.51 if the population consists of 72 ball bearings. 5.53. In Problem 5.51, how many of the random samples would have their means (a) between 22.39 and 22.41 oz, (b) greater than 22.42 oz, (c) less than 22.37 oz, (d) less than 22.38 or more than 22.41 oz 5.54. Certain tubes manufactured by a company have a mean lifetime of 800 hours and a standard deviation of 60 hours. Find the probability that a random sample of 16 tubes taken from the group will have a mean lifetime (a) between 790 and 810 hours, (b) less than 785 hours, (c) more than 820 hours, (d) between 770 and 830 hours. 5.55. Work Problem 5.54 if a random sample of 64 tubes is taken. Explain the difference. 5.56. The weights of packages received by a department store have a mean of 300 lb and a standard deviation of 50 lb. What is the probability that 25 packages received at random and loaded on an elevator will exceed the safety limit of the elevator, listed as 8200 lb Sampling distribution of proportions
5.57. Find the probability that of the next 200 children born, (a) less than 40% will be boys, (b) between 43% and 57% will be girls, (c) more than 54% will be boys. Assume equal probabilities for births of boys and girls. 5.58. Out of 1000 samples of 200 children each, in how many would you expect to find that (a) less than 40% are boys, (b) between 40% and 60% are girls, (c) 53% or more are girls 5.59. Work Problem 5.57 if 100 instead of 200 children are considered, and explain the differences in results. CHAPTER 5 Sampling Theory
5.60. An urn contains 80 marbles of which 60% are red and 40% are white. Out of 50 samples of 20 marbles, each selected with replacement from the urn, how many samples can be expected to consist of (a) equal numbers of red and white marbles, (b) 12 red and 8 white marbles, (c) 8 red and 12 white marbles, (d) 10 or more white marbles 5.61. Design an experiment intended to illustrate the results of Problem 5.60. Instead of red and white marbles, you may use slips of paper on which R or W are written in the correct proportions. What errors might you introduce by using two different sets of marbles 5.62. A manufacturer sends out 1000 lots, each consisting of 100 electric bulbs. If 5% of the bulbs are normally defective, in how many of the lots should we expect (a) fewer than 90 good bulbs, (b) 98 or more good bulbs

