Chapter 04 - Describing Data: Displaying and Exploring Data

Chapter 4

Describing Data: Displaying and Exploring Data

1. A histogram describes a frequency distribution using a series of adjacent rectangles. On the other hand, a dot plot displays a dot for each individual observation. A dot plot may be better if grouping of the data into classes causes a great loss of accuracy.

2. A stem-and-leaf display describes the individual observations similar to a tally. Hence it has slightly more information than a histogram.

3. a. Dot plot

b. 15

c. 7 and 1, respectively

d. 2 and 3

4. a. 19 and 4, respectively

b. 12

5. a. 620 to 629

b. 5

c. 621, 623, 623, 627, 629

6. a. 210 to 219

b. 6

c. 210, 211, 213, 215, 217, 219

7. a. 25

b. one

c. 38, 106

d. 60, 61, 63, 63, 65, 65, 69

e. No values

f. 9

g. 9

h. 76

i. 16

8. a. 50

b. one

c. 126, 270

d. 155, 158, 159

e. No values

f. 13

g. 12

h. 193 or 194

i. 19

9. Stem Leaves

0 5

1 28

3 0024789

4 12366

5 2

There were a total of 16 calls studied. The number of calls ranged from 5 to 52 received. Seven of the 16 subscribers received between 30 and 39 calls.

10. Stem Leaves

3 6

4 7

5 22499

6 0113458

7 035678

8 0344447

9 055

The daily usage ranged from 36 to 95. In a typical day the ATM is used about 70 times.

11. Median = 53 found by (11 + 1)(1/2) therefore 6th value in from lowest.

Q1 = 49 found by (11 + 1)(1/4) therefore 3rd value in from lowest

Q3 = 55 found by (11 + 1)(3/4) therefore 9th value in from lowest

12. Median = 9.53, found by (9.45 + 9.61)/2

Q1 = 7.69 found by 7.59 + (7.99 – 7.59) ¼

Q3 = 12.59 found by 12.22 + (12.71 – 12.22)3/4

13. a. Q1 = 33.25 Q3 = 50.25

b. D2 = 27.8 D8 = 52.6

c. P67 = 47

14. a. Median = 58

b. Q1 = 51.25 Q3 = 66.0

c. D1 = 45.3 D9 = 76.4

d. P33 = 53.53

15. a. 350

b. Q1 = 175 Q3 = 930

c. 755, found by 930 – 175

d. Less than zero, or more than about 2060

e. There are no outliers

f. The distribution is positively skewed

16. a. 450

b. Q1 = 300 Q3 = 700

c. 400, found by 700 – 300

d. Less than zero or more than 1300

e. One outlier at about 1500

f. Distribution is positively skewed

17. The distribution is somewhat positively skewed. Note that dashed line above 35 is longer than below 18.

------

------| + |------

------

----+------+------+------+------+------+---

14 21 28 35 42 49

18. The median is $253. About 25% of the semi-private rooms are less than $214 and 25% above $304. The distribution is negatively skewed.

19. a. The mean is 30.8, found by 154/5. The median is 31.0 and the standard deviation is 3.96, found as.

b. -0.15, found by .

Salary
36 / 1.313131 / 2.264250504
26 / -1.21212 / -1.780894343
33 / 0.555556 / 0.171467764
28 / -0.70707 / -0.353499282
31 / 0.050505 / 0.000128826
0.301453469

0.125 found by

20. a. The mean is 542, found by 8130/15. The median is 546 and the standard deviation is 25.08, found as .

b. –0.478, found by

c. –0.375, found by

21. a. The mean is 21.93, found by 328.9 / 15. The median is 15.8 and the standard deviation is 21.18, found as .

b. 0.868, found by .

c. 2.444, found by

22. a. The mean is $6,744,567, found by $209,081,577 / 31. The median is $1,875,000 and the standard deviation is $8,093,766.

b. 1.805, found by.

c. 1.134, found by

23.

Larger values of x seem to be associated with larger values of y.

24. a. 15

b. More labor hours are required as the number of rooms increases.

25. a. Both variables are nominal.

b. Contingency table

c. Yes, because 32% is more than twice 15%.

26. a. Opinion is ordinal and number of shares is ratio.

b. Contingency table

c. Those who own over 1000 shares.

27. a. Dot plot

b. 15

c. 5

28. Georgetown serves between 8 and 34 patients per day with a typical number being 23. Monks Corners serves between 8 and 36 patients per day with a typical number being 22. Aynor is busier than the other two and serves between 11 and 52 patients per day with a typical number being 30.

29 Stem-and-leaf N = 23

3 3 222

3 3

5 3 77

5 3

10 4 00000

11 4 2

11 4

(6) 4 666666

6 4

6 5

6 5 222222

30 Stem-and-leaf of Year Founded N = 26

Leaf Unit = 10

1 18 1

1 18

2 18 7

5 18 999

7 19 00

11 19 2233

12 19 5

(3) 19 777

11 19 8888888999

1 20 0

Stem-and-leaf of Employees N = 26

Leaf Unit = 10000

(17) 0 00000000000011111

9 0 333

6 0 44

4 0 6

3 0 8

2 1

2 1 45

The distribution of the year founded is relatively symmetric with the peak in the 1970s and 1980s. The distribution of the number of employees is very positively skewed with almost half firms employing less than 10,000 people.

31. a. 84.65, found by (83.7 + 85.6)/2; 68.175, found by 66.6 + (72.9 – 66.6)/4; and 89.425, found by 87.1 + (90.2 –87.1) 3/4.

b. 69.498, found by 66.6 + .46(72.9 – 66.6) and 95.579, found by 93.3 + .43(98.6 – 93.3).

32. a. 133, the 13th ordered value; 89, found by 81 + (97 – 81)/2; and 168, found by
162 + (174 – 162)/2.

b. 283.5, found by 283 + (284 – 283)/2; 189.75, found by 183 + (192 – 183) 3/4; and 492.5, found by 490 + (500 – 490)/4.

c. The typical young adult has more than twice the number of CD’s as the typical senior citizen.

33. a. 31.5, found by 31 + (32- 31)/2; 26.25, found by 25 + (30 – 25)/4; and 35.75, found by
35 + (36 – 35) 3/4.

b. 37.5, found by 37 + (38- 37)/2; 33.25, found by 33 + (34 – 33)/4; and 38.75, found by
38 + (39 – 38) 3/4.

c. Private transportation averages 5 or 6 minutes longer.

34. The distribution is positively skewed. With the first quartile equal to 10 and the third quartile is equal to 40. There are four outliers located at 85, 86, 95 and 99. The median is about 25.

35. The distribution is positively skewed. The first quartile is approximately $20 and the third quartile is approximately $90. There is one outlier located at $255. The median is about 50.

36. a. ------

------+ ------

------

-----|-----|-----|-----|-----|-----|-----|-----|

10 15 20 25 30 35 40 45

b. No outliers

c. The distribution is positively skewed. The median time to change a muffler is 23.50 minutes. The first quartile is 15.75 minutes and the third quartile is 29.25 minutes. The range of time is 10 minutes to 44 minutes.

37. a.

Median is 3733. First quartile is 1478. Third quartile is 6141. So prices over 13,135.5, found by 6141 + 1.5(6141-1478), are outliers. There are three (13925; 20,413 and 44,312).

Median is 0.84. First quartile is 0.515. Third quartile is 1.12. So sizes over 2.0275, found by 1.12 + 1.5(1.12–0.515), are outliers. There are three (2.03, 2.35, and 5.03).

There is a direct association between them. The first observation is larger on both scales.

Shape\Cut / Average / Good / Ideal / Premium / Ultra Ideal / Total
Emerald / 0 / 0 / 1 / 0 / 0 / 1
Marquise / 0 / 2 / 0 / 1 / 0 / 3
Oval / 0 / 0 / 0 / 1 / 0 / 1
Princess / 1 / 0 / 2 / 2 / 0 / 5
Round / 1 / 3 / 3 / 13 / 3 / 23
Total / 2 / 5 / 6 / 17 / 3 / 33

The majority of the diamonds are round (23). Premium cut is most common (17).

The Round Premium combination occurs most often (13).

38. Pearson’s coefficient is 0.4096, found by 3(1322 – 1281.45)/ 297. Software coefficient is 0.3656, found by 8(1.9192)/(8 - 1)(8 - 2).

39. Pearson’s coefficient is –0.2198, found by 3(7.7143 – 8)/ 3.9. Software coefficient is 0.0648, found by 7(0.2779)/(7 - 1)(7 - 2).

40. Old machines cost more to maintain.

41.

The number of accidents appears to decline as the driver ages. The slope is about one fewer accident for each additional four years of driving experience.

42. Older customers tend to order very few condiments and young customers tend to order the largest number.

43. a. 139,340,000, found by adding the four cells.

b. 5.4 % were unemployed, found by 7523/139340

c. 5.6% of the men and 5.1% of the women

44. a.

The first quartile is about 190 and the third is near 250. There are no outliers.

There is a slight direct relationship between size and price.

There is an inverse relationship between distance from the city center and price.

45. a.

There are two outliers: Wrigley Field and Fenway Park.

The first quartile is $65,080,000 and the third is $100,750,000. The distribution is positively skewed, with the New York Yankees a definite outlier.

Higher salaries lead to more wins.

The distribution is fairly uniform between 66 and 96.

46. a. Stem and leaf of wage N = 100, leaf unit = 1000

1 0 9

13 1 011112333334

33 1 55555666677788999999

42 2 000112234

(14) 2 5566888999999

44 3 0001111222233333344

25 3 56779

20 4 14

18 4 5699

14 5 0002

10 5 57

8 6 00

6 6 68

4 7

4 7 5

3 8 333

The values above 60,000 are probably outliers or particularly unusual. The typical value is near 30,000 and 75% of the value are below 35,000.

b. Stem and Leaf of Ed. N = 100, leaf unit = 0.10

1 4 0

2 5 0

3 6 0

4 7 0

8 8 0000

10 9 00

13 10 000

18 11 00000

(43) 12 0000000000000000000000000000000000000000000

39 13 000000

33 14 000000000

24 15 000

21 16 000000000000

9 17 000

6 18 000000

Those who have 8 or fewer “grade school” and those with 18 “post graduate” years of education are outliers. The typical person has 12 and most have between 12 and 14 years of education.

Occupational category 2 has fewer member (5 or 6) and all the others are around 19.

47. a.

The first quartile is 71.5 years and the third is 78.5 years. The distribution is negatively skewed with two outliers (Nigeria and South Africa at 48 and 51).

The first quartile is 8.3 and the third quartile is 24.4. The distribution is symmetric with no outliers.

c. Stem-and-leaf of Cell pho N = 46 Leaf Unit = 1.0

(35) 0 00000000000111111122222222233344444

11 0 68

9 1 123

6 1 5

5 2 0

4 2 7

3 3

3 4

3 5

3 6 3

2 6 59

The distribution is extremely positively skewed. The median is 2 and the mean is around 8, which is above the third quartile of about 5.

4-18