ch7 sampling distribution - suggested problems solutions

STATISTICS FOR SOCIAL SCIENCESBUS 152

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SAMPLING DISTRIBUTIONS

7.2 Sampling Distribution of the Mean

Problem 1 (7.1)SOLUTIONS:

a) P( X < 95) = P(Z < – 2.50) = 0.0062b) P(95 < X < 97.5) = P(– 2.50 < Z < – 1.25) = 0.1056 – 0.0062 = 0.0994c) P( X > 102.2) = P(Z > 1.10) = 1.0 – 0.8643 = 0.1357

d) P( X > A) = P(Z > – 0.39) = 0.65 X = 100 – 0.39(10

25) = 99.22

Problem 2 (7.4)

SOLUTIONS:

a) Sampling Distribution of the Mean for n = 2 (without replacement)

Sample Number Outcomes Sample Means

1 1, 3 = 2

2 1, 6 = 3.5

3 1, 7 = 4

4 1, 9 = 5

5 1, 10 = 5.5

6 3, 6 = 4.5

7 3, 7 = 5

8 3, 9 = 6

9 3, 10 = 6.5

10 6, 7 = 6.5

11 6, 9 = 7.5

12 6, 10 = 8

13 7, 9 = 8

14 7, 10 = 8.5

15 9, 10 = 9.5

Mean of All Possible Sample Means: Mean of All Population Elements:

X

90

156

Both means are equal to 6. This property is called unbiasedness.

b) Sampling Distribution of the Mean for n = 3 (without replacement)


1 1, 3, 6 = 3 1/3

2 1, 3, 7 = 3 2/3

3 1, 3, 9 = 4 1/3

4 1, 3, 10 = 4 2/3

5 1, 6, 7 = 4 2/3

6 1, 6, 9 = 5 1/3

7 1, 6, 10 = 5 2/3

8 3, 6, 7 = 5 1/3

9 3, 6, 9 = 6

10 3, 6, 10 = 6 1/3

11 6, 7, 9 = 7 1/3

12 6, 7, 10 = 7 2/3

13 6, 9, 10 = 8 1/3

14 7, 9, 10 = 8 2/3

15 1, 7, 9 = 5 2/3

16 1, 7, 10 = 6

17 1, 9, 10 = 6 2/3

18 3, 7, 9 = 6 1/3

19 3, 7, 10 = 6 2/3

20 3, 9, 10 = 7 1/3

X

120

206 This is equal to , the population mean.

c) The distribution for n = 3 has less variability. The larger sample size has resulted in sample means being closer to .

d) (a) Sampling Distribution of the Mean for n = 2 (with replacement)


1 1, 1 = 1

2 1, 3 = 2

3 1, 6 = 3.5

4 1, 7 = 4

5 1, 9 = 5

6 1, 10 = 5.5

7 3, 1 = 2

8 3, 3 = 3

9 3, 6 = 4.5

10 3, 7 = 5

11 3, 9 = 6

12 3, 10 = 6.5

13 6, 1 = 3.5

14 6, 3 = 4.5

15 6, 6 = 6

16 6, 7 = 6.5

17 6, 9 = 7.5

18 6, 10 = 8

19 7, 1 = 4

20 7, 3 = 5

21 7, 6 = 6.5

22 7, 7 = 7

23 7, 9 = 8

24 7, 10 = 8.5

25 9, 1 = 5

26 9, 3 = 6

27 9, 6 = 7.5

28 9, 7 = 8

29 9, 9 = 9

30 9, 10 = 9.5

31 10, 1 = 5.5

32 10, 3 = 6.5

33 10, 6 = 8

34 10, 7 = 8.5

35 10, 9 = 9.5

36 10, 10 = 10

Mean of All Possible Mean of AllSample Means: Population Elements:

1 3 6 7 7 12

66

Both means are equal to 6. This property is called unbiasedness.(b) Repeat the same process for the sampling distribution of the mean for n

= 3 (with replacement). There will be different samples.

This is equal to , the population mean.

(c) The distribution for n = 3 has less variability. The larger sample size has resulted in more sample means being close to .

Problem 3 (7.5)

SOLUTIONS:a) Because the population diameter of Ping-Pong balls is approximately

normally distributed, the sampling distribution of samples of 16 will also be approximately normal.

b) P( X < 1.28) = P(Z < – 2.00) = 0.0228c) P(1.31 < X < 1.33) = P(1.00 < Z < 3.00) = 0.99865 – 0.8413 = 0.1574d) P(A < X < B) = P( -0.84 < Z < 0.84) = 0.60

Lower bound: X = 1.30 – 0.84(0.01) = 1.2916Upper bound: X = 1.30 + 0.84(0.01) = 1.3084

Problem 4 (7.7)Time spent using e-mail per session is normally distributed with µ = 8 minutes and = 2 minutes. If you select a random sample of 25 sessions,a. what is the probability that the sample mean is between 7.8 and 8.2 minutes?b. what is the probability that the sample mean is between 7.5 and 8 minutes?c. if you select a random sample of 100 sessions, what is the probability that the sample mean is between 7.8 and 8.2?d. explain the difference in the results of (a) and (c).

SOLUTIONS:

a) X =

n

2

250.4

P(7.8 < X < 8.2) = P(– 0.50 < Z < 0.50) = 0.6915 – 0.3085 = 0.3830b) P(7.5 < X < 8.0) = P(– 1.25 < Z < 0) = 0.5 – 0.1056 = 0.3944c) P(7.8 < X < 8.2) = P(– 1.00 < Z < 1.00) = 0.8413 – 0.1587 = 0.6826d) With the sample size increasing from n = 25 to n = 100, more sample

means will be closer to the distribution mean. The standard error of the sampling distribution of size 100 is much smaller than that of size 25, so the likelihood that the sample mean will fall within 0.2 minutes of the mean is much higher for samples of size 100 (probability = 0.6826) than for samples of size 25 (probability = 0. 3830).

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7.4 Types of Survey Sampling

Problem 5 (7.26)SOLUTION

A simple random sample would be less practical for personal interviews because of travel costs (unless interviewees are paid to attend a central interviewing location).


This is a random sample because the selection is based on chance. It is not a simple random sample because A is more likely to be selected than B or C.


Here all members of the population are equally likely to be selected and the sample selection mechanism is based on chance. But not every sample of size 2 has the same chance of being selected. For example the sample “B and C” is impossible.


(a) Since a complete roster of full-time students exists, a simple random sample of 200 students could be taken. If student satisfaction with the quality of campus life randomly fluctuates across the student body, a systematic 1-in-20 sample could also be taken from the population frame. If student satisfaction with the quality of life may differ by gender and by experience/class level, a stratified sample using eight strata, female freshmen through female seniors and male freshmen through male seniors, could be selected. If student satisfaction with the quality of life is thought to fluctuate as much within clusters as between them, a cluster sample could be taken.

(b) A simple random sample is one of the simplest to select. The population frame is the registrar’s file of 4,000 student names.

(c) A systematic sample is easier to select by hand from the registrar’s records than a simple random sample, since an initial person at random is selected and then every 20th person thereafter would be sampled. The systematic sample would have the additional benefit that the alphabetic distribution of sampled students’ names would be more comparable to the alphabetic distribution of student names in the campus population.

(d) If rosters by gender and class designations are readily available, a stratified sampleshould be taken. Since student satisfaction with the quality of life may indeed differ by gender and class level, the use of a stratified sampling design will not only ensure all strata are represented in the sample, it will also generate a more representative sample and produce estimates of the population parameter that have greater precision.

(e) If all 4,000 full-time students reside in one of 20 on-campus residence halls which fully integrate students by gender and by class, a cluster sample should be taken. A cluster could be defined as an entire residence hall, and the students of a single randomly selected residence hall could be sampled. Since the dormitories are fully integrated by floor, a cluster could alternatively be defined as one floor of one of the 20 dormitories. Four floors could be randomly sampled to produce the required 200 student sample. Selection of an entire dormitory may make distribution and collection of the survey easier to accomplish. In contrast, if there is some variable other than gender or class that differs across dormitories, sampling by floor may produce a more representative sample.


(a) A stratified sample should be taken so that each of the four strata will be proportionately represented.

(b) Since the stratum may differ in the invoice amount, it may be more important to sample a larger percentage of invoices in stratum 1 and

stratum 2, and smaller percentages in stratum 3 and stratum 4. For example, 50/5000 = 1% so 1% of 500 = 5 invoices should be selected from stratum 1; similarly 10% = 50 should be selected from stratum 2, 20% = 100 from stratum 3, and 69% = 345 from stratum 4.

(c) It is not simple random sampling because, unlike the simple random sampling, it ensures proportionate representation across the entire population.

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7.5 Evaluating Survey Worthiness


Before accepting the results of a survey of college students, you might want to know, for example:

Who funded the survey? Why was it conducted?What was the population from which the sample was selected?What sampling design was used?What mode of response was used: a personal interview, a telephone

interview, or a mail survey? Were interviewers trained? Were survey questions field-tested?

What questions were asked? Were they clear, accurate, unbiased, valid?What operational definition of “vast majority” was used?What was the response rate?What was the sample size?


(a) Possible coverage error: Only employees in a specific division of the company were sampled.

(b) Possible nonresponse error: No attempt is made to contact nonrespondents to urge them to complete the evaluation of job satisfaction.

(c) Possible sampling error: The sample statistics obtained from the sample will not be equal to the parameters of interest in the population.

(d) Possible measurement error: Ambiguous wording in questions asked on the questionnaire.


Before accepting the results of the survey on AOL subscribers, you might want to know, for example:

Who funded the study? Why was it conducted?What was the population from which the sample was selected?What sampling design was used?What mode of response was used: a personal interview, a telephone


What other questions were asked? Were they clear, accurate, unbiased, and valid?What was the response rate?


Before accepting the results of the survey conducted by Forrester Research Inc., you might want to know, for example:

Who funded the study? Why was it conducted?What was the population from which the sample was selected?

What was the sample size?What sampling design was used?What mode of response was used: a personal interview, a telephone


What other questions were asked? Were they clear, accurate, unbiased, and valid?What was the response rate?


Before accepting the results of the Maritz poll, you might want to know, for example:Who funded the study? Why was it conducted?What was the population from which the sample was selected?What sampling design was used?What mode of response was used: a personal interview, a telephone


What operational definitions of "often or sometimes eating or drinking while driving" and “talking on cellphone” were used?What other questions were asked? Were they clear, accurate, unbiased, and valid?What was the response rate?


Before accepting the results of the survey conducted by MessageOne, you might want to know, for example:

Who funded the study? Why was it conducted?What was the population from which the sample was selected?What was the sample size?What sampling design was used?What mode of response was used: a personal interview, a telephone


What operational definitions of "can’t live without it", “important”, “not essential”, “don’t use” and “not important” were used?What other questions were asked? Were they clear, accurate, unbiased, and valid?What was the response rate?


Before accepting the results of the survey conducted by Caravan, you might want to know, for example:

Who funded the study? Why was it conducted?What was the population from which the sample was selected?What was the sample size?What sampling design was used?What mode of response was used: a personal interview, a telephone


What operational definitions of "great food", “reasonable prices”, “atmosphere”, “quick service” and “don’t know” were used?What other questions were asked? Were they clear, accurate, unbiased, and valid?What was the response rate?

ch7 sampling distribution - suggested problems solutions

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