2-2. Exercises

1. Basic Exercises

Ex 1. (1, 2, 6) 표본 데이터 세트에 대하여 다음을 구하라.

1. $\Sigma x$

2. $\Sigma x^2$

3. $\Sigma (x - 3)$

4. $\Sigma (x-3)^2$

Ex 2. (-1, 0, 1, 4) 데이터 세트에 대하여 다음을 구하라.

1. $\Sigma x$

2. $\Sigma x^2$

3. $\Sigma (x - 1)$

4. $\Sigma (x-1)^2$

Ex 3. 다음 데이터 세트의 평균(mean), 중앙값(median)과 최빈값(mode)을 구하라.

1 2 3 4

Ex 4. 다음 데이터 세트의 평균(mean), 중앙값(median)과 최빈값(mode)을 구하라.

3 3 4 4

Ex 5. 다음 데이터 세트의 평균(mean), 중앙값(median)과 최빈값(mode)을 구하라.

2 1 2 7

Ex 6. 다음 데이터 세트의 평균(mean), 중앙값(median)과 최빈값(mode)을 구하라.

-1 0 1 4 1 1

Ex 7. 다음의 데이블 형태로 표현된 표본의 균(mean), 중앙값(median)과 최빈값(mode)을 구하라.

x   1  2  7
f   1  2  1

Ex 8. 다음의 데이블 형태로 표현된 표본의 균(mean), 중앙값(median)과 최빈값(mode)을 구하라.

x   -1  0  1  4
f    1  1  3  1

Ex 9. 평균이 중앙값보다 더 큰 표본 크기 3인 데이터 세트를 작성하라.

Ex 10. 평균이 중앙값보다 더 작은 표본 크기 3인 데이터 세트를 작성하라.

Ex 11. 평균과 중앙값과 최빈값이 다 같은 표본 크기 4인 데이터 세트를 작성하라.

Ex 12. 중앙값과 최빈값이 같지만 평균값이 다른 표본 크기 4인 데이터 세트를 작성하라.

2. Applications Exercises

Ex 13. 다음 데이터 세트의 평균과 중앙값을 구하라.

132 162 133 145 148 139 147 160 150 153

Ex 14. 다음 데이터 세트의 평균과 중앙값을 구하라.

127 152 138 110 152 113 131 148 135 158

Ex 15. 52 세대의 조사를 통해 가구당 보유 차량 대수의 평균값, 중앙값 그리고 최빈값은 구하라.

x   0  1  2  3  4  5  6  7
f   2 12 15 11  6  3  1  2

Ex 16. 아침 출근 시간에 무작위로 120대의 차량을 대상으로 관측한 탑승객의 수이다. 평균, 중앙값, 그리고 최빈값을 구하라.

x   1   2   3   4   5
f  84  29   3   3   1

Ex 17. 16d 규격의 못이 담겨 있는 1파운드(450그램) 짜리 25상자를 무작위로 선택하여, 각 상자에 담겨있는 못의 갯수를 세어 본 결과가 다음과 같다. 평균, 중앙값, 그리고 최빈값을 구하라.

x 47 48 49 50 51
f  1  3 18  2  1

3. Additional Exercises

Ex 17. Five laboratory mice with thymus leukemia are observed for a predetermined period of 500 days. After 500 days, four mice have died but the fifth one survives. The recorded survival times for the five mice are

493 421 222 378 500*

where $500*$ indicates that the fifth mouse survived for at least 500 days but the survival time (i.e., the exact value of the observation) is unknown.

1. Can you find the sample mean for the data set? If so, find it. If not, why not?

2. Can you find the sample median for the data set? If so, find it. If not, why not?

Ex 18. Five laboratory mice with thymus leukemia are observed for a predetermined period of 500 days. After 450 days, three mice have died, and one of the remaining mice is sacrificed for analysis. By the end of the observational period, the last remaining mouse still survives. The recorded survival times for the five mice are

222 421 378 450* 500* 

where * indicates that the mouse survived for at least the given number of days but the exact value of the observation is unknown.

1. Can you find the sample mean for the data set? If so, find it. If not, explain why not.

2. Can you find the sample median for the data set? If so, find it. If not, explain why not.

Ex 19. A player keeps track of all the rolls of a pair of dice when playing a board game and obtains the following data.

x    2   3   4   5   6   7  8   9   10  11  12
f   10  29  40  56  68  77 67  55   39  28  11

Find the mean, the median, and the mode.

Ex 20. Cordelia records her daily commute time to work each day, to the nearest minute, for two months, and obtains the following data.

x   26   27   28   29   30   31   32
f    3    4   16   12    6    2    1

1. Based on the frequencies, do you expect the mean and the median to be about the same or markedly different, and why?

2. Compute the mean, the median, and the mode.

Ex 21. An ordered stem and leaf diagram gives the scores of 71 students on an exam.

10 |  00
 9 |  111123
 8 |  01122345
 7 |  8897000112445666777889
 6 |  012223445777788
 5 |  02334467789
 4 |  25688
 3 |  99

1. Based on the shape of the display, do you expect the mean and the median to be about the same or markedly different, and why?

2. Compute the mean, the median, and the mode.

Ex 22. A man tosses a coin repeatedly until it lands heads and records the number of tosses required. (For example, if it lands heads on the first toss he records a 1; if it lands tails on the first two tosses and heads on the third he records a 3.) The data are shown.

x      1     2    3    4    5    6  7  8  9  10
f    384   208   98   56   28   12  8  2  3   1

1. Find the mean of the data.

2. Find the median of the data.

Ex 23.

1. Construct a data set consisting of ten numbers, all but one of which is above average, where the average is the mean.

2. Is it possible to construct a data set as in part (a) when the average is the median? Explain.

Ex 24. Show that no matter what kind of average is used (mean, median, or mode) it is impossible for all members of a data set to be above average.

Ex 25.

1. Twenty sacks of grain weigh a total of 1,003 lb. What is the mean weight per sack?

2. Can the median weight per sack be calculated based on the information given? If not, construct two data sets with the same total but different medians.

Ex 26. Begin with the following set of data, call it Data Set I.

5 −2 6 14 −3 0 1 4 3 2 5

1. Compute the mean, median, and mode.

2. Form a new data set, Data Set II, by adding 3 to each number in Data Set I. Calculate the mean, median, and mode of Data Set II.

3. Form a new data set, Data Set III, by subtracting 6 from each number in Data Set I. Calculate the mean, median, and mode of Data Set III.

4. Comparing the answers to parts (a), (b), and (c), can you guess the pattern? State the general principle that you expect to be true.

4. LARGE DATA SET EXERCISES

Note: All of the data sets associated with these questions are missing, but the questions themselves are included here for reference.

65KB

data1.xls

Data Set 1

17KB

data7.xls

Data Set 7

12KB

data7A.xls

Data Set 7A

13KB

data7B.xls

Data Set 7b

1. Large Data Set 1 lists the SAT scores and GPAs of 1,000 students.

   1. Compute the mean and median of the 1,000 SAT scores.

   2. Compute the mean and median of the 1,000 GPAs.

2. Large Data Set 1 lists the SAT scores of 1,000 students.

   1. Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean $μ$ .

   2. Regard the first 25 observations as a random sample drawn from this population. Compute the sample mean $\bar{x}$ and compare it to $μ$ .

   3. Regard the next 25 observations as a random sample drawn from this population. Compute the sample mean $\bar{x}$ and compare it to $μ$ .

3. Large Data Set 1 lists the GPAs of 1,000 students.

   1. Regard the data as arising from a census of all freshman at a small college at the end of their first academic year of college study, in which the GPA of every such person was measured. Compute the population mean μ.

   2. Regard the first 25 observations as a random sample drawn from this population. Compute the sample mean $\bar{x}$ and compare it to $μ$ .

   3. Regard the next 25 observations as a random sample drawn from this population. Compute the sample mean $\bar{x}$ and compare it to $μ$ .

4. Large Data Sets 7, 7A, and 7B list the survival times in days of 140 laboratory mice with thymic leukemia from onset to death.

   1. Compute the mean and median survival time for all mice, without regard to gender.

   2. Compute the mean and median survival time for the 65 male mice (separately recorded in Large Data Set 7A).

   3. Compute the mean and median survival time for the 75 female mice (separately recorded in Large Data Set 7B).