CHAPTER 9 SAMPLING DISTRIBUTIONS MULTIPLE CHOICE …

[Pages:20]CHAPTER 9 SAMPLING DISTRIBUTIONS

MULTIPLE CHOICE QUESTIONS

In the following multiple choice questions, pleas e circle the correct answer.

1.

As a general rule, the normal distribution is used to approximat e the

sa m pling distribution of the sam ple proportion only if

a. the sample size n is greater than 30

b. the population proportion p is close to 0.50

c. the underlying population is normal

d. np and n(1p) are both greater than 5

ANSWER: d

2.

Random samples of size 49 are taken from an infinite population whose

mean is 300 and stand ard deviation is 21. The mean and standard error of

the sample mean, respectively, are:

a. 300 and 21

b. 300 and 3

c. 70 and 230

d. 49 and 21

ANSWER: b

3.

A normally distributed population with 200 element s has a mean of 60 and a

stand ard deviation of 10. The probability that the mean of a sample of 25

eleme nts taken from this population will be smaller than 56 is

a. 0.0166

b. 0.0228

c. 0.3708

d. 0.0394

ANSWER: a

4.

Given an infinite population with a mean of 75 and a stand ar d deviation of

12, the probability that the mean of a sample of 36 observations, taken at

random from this population, exceeds 78 is

a. 0.4332

b. 0.0668

c. 0.0987

d. 0.9013

ANSWER: b

5.

A population that consists of 500 observations has a mean of 40 and a

standard deviation of 15. A sample of size 100 is taken at random from this

population. The stand ard error of the sample mean equals:

a. 2.50

139

140 Chapter Nine

b. 12.50 c. 1.343 d. 1.50 ANSWER: c

6.

An infinite population has a mean of 60 and a standar d deviation of 8. A

sample of 50 observations will be taken at rando m from this population. The

probability that the sam ple mean will be betwe e n 57 and 62 is

a. 0.9576

b. 0.9960

c. 0.2467

d. 0.3520

ANSWER: a

7.

If all possible sa m pl e s of size n are drawn from an infinite population with a

mean of 15 and a standard deviation of 5, then the stand ard error of the

sample mean equals 1.0 only for samples of size

a. 5

b. 15

c. 25

d. 75

ANSWER: c

8.

If the stand a r d error of the sam pling distribution of the sa m pl e proportion is

0.0229 for samples of size 400, then the population proportion must be

either

a. 0.4 or 0.6

b. 0.5 or 0.5

c. 0.2 or 0.8

d. 0.3 or 0.7

ANSWER: d

9.

As a general rule in computing the stand ard error of the sample mean, the

finite population correction factor is used only if the :

a. sample size is smaller than 10% of the population size

b. population size is smaller than 10% of the sample size

c. sample size is greater than 1% of the population size

d. population size is greater than 1% of the sample size

ANSWER: c

10. Given that X is a bino mi al rando m variable, the binomi al prob a bility P(X x )

is approximat e d by the area under a norm al curve to the right of a. x 0.5 b. x+0.5 c. x 1 d. x+1

Sampling Distributions 141 ANSWER: a 11. Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals: a. 12.649 b. 25.0 c. 2.56 d. 3.125 ANSWER: d 12. A sam ple of size 200 will be taken at rando m from an infinite population. Given that the population proportion is 0.60, the probability that the sam ple proportion will be greater than 0.58 is a. 0.281 b. 0.719 c. 0.580 d. 0.762 ANSWER: b 13. A sample of size 40 will be taken from an infinite population whose mean and stand ard deviation are 68 and 12, respectively. The probability that the sample mean will be larger than 70 is a. 0.3970 b. 0.4332 c. 0.1469 d. 0.0668 ANSWER: c

14. A sample of size n is select ed at random from an infinite population. As n increas e s, which of the following state m e n t s is true? a. The population standard deviation decrease s b. The standard error of the sample mean decreas e s c. The population standard deviation increases d. The standard error of the sample mean increas e s ANSWER: b

15. The finite population correction factor should not be used when: a. we are sampling from an infinite population b. we are sam pling from a finite popul ation c. sample size is greater than 1% of the population size d. None of the above state m e n t s is correct ANSWER: a

142 Chapter Nine 16. If the stand a r d error of the sam pling distribution of the sa m pl e proportion is

0.0337 for samples of size 200, then the population proportion must be either: a. 0.25 b. 0.75 c. 0.20 or 0.80 d. 0.35 or 0.65 e. 0.30 or 0.70 ANSWER: c 17. Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are: a. 9 and 45 b. 45 and 9 c. 81 and 45 d. 45 and 1 ANSWER: d 18. A sample of 250 observations will be select ed at random from an infinite population. Given that the population proportion is .25, the stan d a r d error of the sampling distribution of the sample proportion is : a. 0.0274 b. 0.50 c. 0.0316 d. 0.0548 ANSWER: a

19. A sam pl e of size 25 is select e d at rando m from a finite popul ation. If the finite population correction factor is 0.632 5, then the population size is: a. 10 b. 41 c. 15 d. 35 ANSWER: b

20. If two popul ations are norm ally distribut e d, the sam pling distribution of the sample mea n difference X1 - X 2 will be: a. approximat ely normally distributed b. normally distribut e d only if both sam ple sizes are great er than 30 c. normally distributed d. normally distribut e d only if both population sizes are great er than 30 ANSWER: c

Sampling Distributions 143

21. Two samples are selected at random from two independ e n t normally distributed populations. Sample 1 has 49 observations and has a mean of 10 and a standard deviation of 5. Sample 2 has 36 observations and has a mean of 12 and a standard deviation of 3. The standard error of the

sampling distribution of the sam ple mean difference X1 - X 2 is a. 0.1853 b. 0.7602 c. 0.7331 d. 0.8719 ANSWER: d

22. Given a binomial distribution with n trials and probability p of a success on any trial, a conventional rule of thumb is that the norm al distribution will provide an adeq u a t e approxim a tion of the binomial distribution if

a. np 5 and n(1 p) 5 b. np 5 and n(1 p) 5 c. np 5 and n(1 p) 5 d. np 5 and n(1 p) 5

ANSWER: a

23. If two rando m sa m pl e s of sizes n1 and n2 are selec t e d inde p e n d e n tl y from

two popula tion s with me a n s ?1 and ?2 , then the me a n of the sa m pling

distribution of the sam ple mean difference, X1 - X 2 , equals:

a. ?1 + ?2 b. ?1 ?2 c. ?1 / ?2 d. ?1 ?2

ANSWER: b

24. If two rando m sa m pl e s of sizes n1 and n2 are selec t e d inde p e n d e n tl y from

two populations

with variances

2 1

a

n

d

2 2

,

then

the

st a n d a r d

error of the

sampling distribution of the sam ple mean difference, X1 - X 2 , equals:

a.

(

2 1

-

2 2

)

/

n1n

2

b.

(

2 1

+

2 2

)/

n1n2

c.

2 1

n1

-

2 2

n2

d.

2 1

n1

+

2 2

n2

ANSWER: d

25. Suppose that the probability p of a success on any trail of a binomial distribution equals 0.90. Then for which of the following number of trials, n,

144 Chapter Nine

would the normal distribution provide a good approximation to the binomial distribution? a. 25 b. 35 c. 45 d. 55 ANSWER: d 26. If two rando m sa m pl e s of sizes n1 and n2 are selec t e d inde p e n d e n tl y from two non normally distributed populations, then the sampling distribution of the sample mea n difference, X1 - X 2 , is a. always non normal b. always normal c. approxi m a t el y norm al only if n1 and n2 are both larg er tha n 30 d. approxi m a t el y norm al reg ar dl e s s of n1 and n2 ANSWER: c 27. Given that X is a binomi al rando m variabl e, the bino mi al prob a bility P(X= x ) is approximated by the area under a normal curve between a. x 0.5 and 0.0 b. 0.0 and x+0.5 c. 1x and 1+ x d. x 0.5 and x +0.5 ANSWER: d

28. The Centr al Limit Theor e m stat e s that, if a rando m sa m pl e of size n is drawn from a population, then the sampling distribution of the sample mean X : a. is approxim a t ely norm al if n > 30 b. is approxim a t ely norm al if n < 30 c. is approxim a t ely norm al if the und erlying population is norm al d. has the same variance as the population ANSWER: a

29. The expecte d value of the sampling distribution of the sample mean X

equals the population mean ? :

a. when the population is normally distributed b. when the population is symmetric c. when the population size N > 30 d. for all populations ANSWER: d

Sampling Distributions 145 30. If all possible sa m pl e s of size n are drawn from an infinite population with a

mean of ? and a standard deviation of , then the standard error of the

sample mean is inversely proportional to:

a. ?

b.

c. n d. n ANSWER: d

31. Given that X is a bino mi al rando m variable, the binomi al prob a bility P(X x )

is approximat e d by the area under a normal curve to the left of a. x b. ?x c. x +0.5 d. x 0.5 ANSWER: c 32. The standard deviation of the sampling distribution of the sample mean is also called the: a. central limit theor e m b. standard error of the mean c. finite population correction factor d. population standard deviation ANSWER: b

33. If a rando m sa m pl e of size n is drawn from a nor m al population, then the sampling distribution of the sam ple mean X will be: a. normal for all values of n b. normal only for n > 30 c. approximat ely normal for all values of n d. approximat ely normal only for n > 30 ANSWER: a

34. If all possible sam pl e s of size n are drawn from a population, the proba bility distribution of the sample mean X is called the: a. stand ard error of X b. expecte d value of X c. sampling distribution of X d. normal distribution ANSWER: c TRUE/FALSE QUESTIONS

146 Chapter Nine

35. In an effort to identify the true proportion of college freshm a n who are under 18 year s of age, a rando m sam pl e of 500 fresh m a n was take n. Only fifty of them were under the age of 18. The value 0.10 would be used as a point estimat e to the true proportion of under age 18 freshm a n. ANSWER: T

36. The central limit theor e m is basic to the conce p t of statistical inferenc e, because it permits us to draw conclusions about the population based strictly on sample data, and without having any knowledge about the distribution of the underlying population. ANSWER: T

37. When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample mea n s will be norm al regardless of sam pl e size n . ANSWER: T

38. The me a n of the sam pling distribution of the sa m pl e proportion p^ , when the sample size n = 100 and the population proportion p = 0.92, is 92.0. ANSWER: F

39. The stand a r d error of the sam pling distribution of the sa m pl e proportion p^ , when the sample size n = 100 and the population proportion p = 0.30, is 0.0021. ANSWER: F

40. Recall the rule of thumb used to indicate when the normal distribution is a good approxi m a tio n of the sa m pling distribution for the sa m pl e proportion p^ . For the combination n = 50; p = 0.05, the rule is satisfied. ANSWER: F

41. The standard error of the mean is the standard deviation of the sampling distribution of X . ANSWER: T

42. The standard deviation of the sampling distribution of the sample mean is also called the centr al limit theor e m . ANSWER: F

43. Consider an infinite population with a mean of 100 and a standard deviation of 20. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals 2.5. ANSWER: T

44. If all possible sa m pl e s of size n are drawn from an infinite population with a mean of 60 and a standard deviation of 8, then the stand ard error of the sample mean equals 1.0 only for samples of size 64. ANSWER: T

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