Chapter 7 Full Solutions



4 Measures of Dispersion

Review Exercise 4 (p. 4.5)

|1. |(a) |Amount ($) |Class mark ($) |Frequency |

| | |50.5 ( 149.5 |100 |4 |

| | |150.5 ( 249.5 |200 |10 |

| | |250.5 – 349.5 |300 |7 |

| | |350.5 – 449.5 |400 |14 |

| | |450.5 – 549.5 |500 |5 |

(b) Class interval $350.5 − $449.5 has the highest frequency.

(c) The required percentage of students [pic]

2. (a) From the graph, 14 programmes have the viewing rate of 10% or below.

∴ 14 programmes will be subject to review.

(b) For the programmes which have more than 1 million viewers, the viewing rate is greater than [pic]i.e. 20%.

From the graph, 46 programmes have the viewing rate of 20% or below.

∴ Number of programmes having more than

1 million viewers

[pic]

3. (a) Mean income[pic]

Arrange the monthly incomes in ascending order,

i.e. $6200, $6700, $7400, $7600, $8000, $8500, $9200, $9600, $25 700

Median income[pic]

(b) The median can better represent the average monthly income of the employees since the extreme data

($25 700) will greatly affect the value of the mean.

4. (a) From the graph, the median score ’ 57

[pic]

(b) From the graph, the median score ’[pic]

[pic]

5. (a) Mean of data in set Y

[pic]

Let x be the datum added in set Y.

[pic]

∴ The required datum is 18.

(b) Arrange the set of data in ascending order,

i.e. (10, (2, 4, 4, 5, 12, 13, 20, 26.

Median ’ 5

If the median remains unchanged, the required datum should equal to the median.

∴ The required datum is 5.

Activity

Activity 4.1 (p. 4.7)

|1. | |Mean |Median |Mode |

| |Athlete A |7.5 |7.5 |7.5 |

| |Athlete B |7.5 |7.5 |7.5 |

2. Since the measures of central tendency for the two athletes are all the same, they performed equally well.

3. From the dot-plots, we observe that the scores of athlete A are more spread out.

Hence, the performance of athlete B is more stable.

Activity 4.2 (p. 4.27)

|1. |(a) | |Set A |Set B |

| | |Range |9 |9 |

| | |Inter-quartile range |5 |5 |

b) No

2. No. By inspection, set A is more dispersed.

Activity 4.3 (p. 4.40)

1. English

2. (a) 6

(b) 5

3. No. We don’t know the distributions of the marks./ We don’t know how many students got higher marks than Mary in each subject. (or any other reasonable answers)

4. The dispersion of the marks of the class in each subject.

Activity 4.4 (p. 4.53)

|1. |Measure of |Data set A |Data set B |Data set C |

| |dispersion | | | |

| |Range |3 |3 |6 |

| |Inter-quartile |1.5 |1.5 |3 |

| |range | | | |

| |Standard deviation|1 |1 |2 |

2. The measures of dispersion of data set B are the same as those of data set A.

3. The measures of dispersion of data set C are twice those of data set A.

Let’s Discuss

Let’s Discuss (p. 4.30)

No, the result of their suggested formula is always zero.

Let’s Discuss (p. 4.55)

(a) 40, 90 (or any other reasonable answers)

(b) 65, 66 (or any other reasonable answers)

Classwork

Classwork (p. 4.12)

| |Data |Lower half of the |Upper half of the |

| | |data |data |

|(a) |2, 5, 6, 8, 12, 13 |2, 5, 6 |8, 12, 13 |

|(b) |1, 5, 7, 9, 11, 14, |1, 5, 7 |11, 14, 15 |

| |15 | | |

|(c) |(3, (1, 0, 2, 2, 4, |(3, (1, 0, 2 |2, 4, 6, 7 |

| |6, 7 | | |

|(d) |18, 30, 10, 34, 45, |8, 10, 18, 18 |28, 30, 34, 45 |

| |18, 24, 8, 28 | | |

|(e) |(4, (4, (1, 0, (1, |(5, (4, (4, (3, (2|(2, (1, (1, 0, 0 |

| |0, (3, (2, (5, (2 | | |

| |Data |Q1 |Q3 |IQR |

| | | | |(Q3 ( Q1) |

|(a) |2, 5, 6, 8, 12, 13 |5 |12 |7 |

|(b) |1, 5, 7, 9, 11, 14, |5 |14 |9 |

| |15 | | | |

|(c) |(3, (1, 0, 2, 2, 4, |(0.5 |5 |5.5 |

| |6, 7 | | | |

|(d) |18, 30, 10, 34, 45, |14 |32 |18 |

| |18, 24, 8, 28 | | | |

|(e) |(4, (4, (1, 0, (1, |(4 |(1 |3 |

| |0, (3, (2, (5, (2 | | | |

Classwork (p. 4.20)

(a) Median of A ’ median of B

Range of A > range of B

IQR of A < IQR of B

(b) Median of A > median of B

Range of A < range of B

IQR of A ’ IQR of B

Classwork (p. 4.29)

(a) [pic]

( = 3.08

(b) [pic]

( =[pic]

(c) [pic]

( =[pic]

(d) [pic]

( =[pic]

Classwork (p. 4.31)

| |Data |( (cor. to 3 sig. |

| | |fig.) |

|(a) |{69, 72, 65, 63, 59, 56, 52, 52} |7.04 |

|(b) |{68, 71, 58, 42, 48, 52, 53, 51} |9.22 |

|(c) |{64, 63, 70, 72, 57, 69, 60, 71} |5.19 |

|(d) |{40, 40, 40, 21, 21, 21, 55, 55, 55, 55} |14.1 |

Classwork (p. 4.36)

(a) It is easy to calculate.

It is easy to understand by most people.

The maximum and the minimum temperatures in a day are important data for people.

(b) It takes all data into account.

(c) The inter-quartile range can be conveniently found from a cumulative frequency polygon.

It is not affected by extreme values.

(d) It is not affected by extreme values.

Classwork (p. 4.42)

1. (a) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

2. (a) Test 1

(b) Test 3

Classwork (p. 4.45)

| | |Interval |Percentage |

| | | |of data |

|(a) |[pic] |between[pic] |[pic] |

| | |and[pic] | |

|(b) |[pic] |between[pic] |[pic] |

| | |and[pic] | |

|(c) |[pic] |greater |[pic] |

| | |than[pic] | |

Classwork (p. 4.55)

For the 1st row: [pic]

[pic]

[pic] [pic]

For the 2nd row: [pic]

For the 3rd row: [pic]

For the 4th row: [pic]

Quick Practice

Quick Practice 4.1 (p. 4.9)

a) Range of salaries of team A

[pic]

Range of salaries of team B

[pic]

(b) Since the range of salaries of team A > the range of salaries of team B, the salaries of team A are more dispersed.

Quick Practice 4.2 (p. 4.10)

Range of waiting times at Airline X check-in counter

’ (20.5 – 0.5) min

’ 20 min

Range of waiting times at Airline Y check-in counter

’ (30.5 – 0.5) min

’ 30 min

Since the range of waiting times at Airline X check-in counter

< the range of waiting times at Airline Y check-in counter, the waiting times at Airline X check-in counter are less dispersed.

Quick Practice 4.3 (p. 4.13)

Inter-quartile range of set A

’ 23 – 15

’ 8

Inter-quartile range of set B

[pic]

∵ IQR for set B > IQR for set A

∴ Based on the inter-quartile ranges, set B has a greater dispersion.

Quick Practice 4.4 (p. 4.14)

(a) ∵ The lower quartile is the value corresponding to the cumulative frequency of 50.

∴ From the graph, the lower quartile[pic]

∵ The upper quartile is the value corresponding to the cumulative frequency of 150.

∴ From the graph, the upper quartile[pic]

(b) Inter-quartile range[pic]

Quick Practice 4.5 (p. 4.20)

∵ Smallest datum ’ 13

Largest datum ’ 82

Median ’ 28

Lower quartile ’ 18

Upper quartile ’ 56

∴ The required box-and-whisker diagram is:

[pic]

Quick Practice 4.6 (p. 4.21)

(a) Range of the data set[pic]

IQR of the data set[pic]

(b) ∵ Upper quartile ’ 195

∴ The total number of data[pic]

Quick Practice 4.7 (p. 4.22)

a) The minimum, median, upper quartile and maximum of the number of visitors to the Modern Art Gallery are all higher than that to the Chinese Art Gallery. Therefore, the Modern Art Gallery has more visitors in general.

b) Since the box for the Chinese Art Gallery is shorter, the inter-quartile range of the number of visitors to the Chinese Art Gallery is smaller. Hence, the Chinese Art Gallery has a more stable number of visitors.

c) No. The lower-quartile of the number of visitors to the Modern Art Gallery is lower than that to the Chinese Art Gallery. Therefore, there are at least 25% of the days in which the visitors to the Chinese Art Gallery are more than that to the Modern Art Gallery.

Quick Practice 4.8 (p. 4.32)

a) Standard deviation of the height cleared by Athlete X ((X)

[pic]

Standard deviation of the height cleared by Athlete Y ((Y)

[pic]

b) Since (Y < (X, the heights cleared by Athlete Y are less dispersed. Therefore, Athlete Y has a more stable performance.

Quick Practice 4.9 (p. 4.33)

Standard deviation of the number of countries visited

[pic]

Quick Practice 4.10 (p. 4.34)

Standard deviation of the distances[pic](cor. to 3 sig. fig.)

Quick Practice 4.11 (p. 4.34)

(a) Standard deviation of the salaries of the employees in Company A

[pic](cor. to 3 sig. fig.)

Standard deviation of the salaries of the employees in Company B

[pic](cor. to 3 sig. fig.)

c) Since the standard deviation of the salaries of the employees in Company B is smaller, Company B offers more uniform salaries.

Quick Practice 4.12 (p. 4.43)

Standard score of the height cleared by Jack

[pic]

Standard score of the height cleared by Mary

[pic]

Therefore, the height cleared by Jack is 0.938 standard deviation above the mean and the height cleared by Mary is 0.435 standard deviation above the mean. Jack’s recorded height is relatively higher in his own group.

Quick Practice 4.13 (p. 4.43)

(a) For practice 1,

mean ’ 30.5,

standard deviation ’ 3.2404 (cor. to 4 d.p.)

John’s standard score[pic]

For practice 2,

mean ’ 19.625,

standard deviation ’ 2.6897 (cor. to 4 d.p.)

John’s standard score[pic](cor. to 3 sig. fig.)

(b) ∵ Standard score in practice 1 > standard score in practice 2

∴ John performs better in practice 1.

Quick Practice 4.14 (p. 4.47)

(a) ∵ 167 cm ’ [175 – 2(4)] cm ’[pic]

183 cm ’ [175 + 2(4)] cm ’[pic]

∴ The required percentage is 95%.

b) The required number of players

[pic]

Quick Practice 4.15 (p. 4.48)

(a) ∵ 45 min ’ [35 + 2(5)] min ’[pic]

∴ The percentage of delivery times greater than 45 min

[pic]

∴ Number of food coupons given out

[pic]

(b) ∵ 45 min ’ [30 + 3(5)] min ’[pic]

∴ The percentage of delivery times greater than 45 min

[pic]

Quick Practice 4.16 (p. 4.56)

(a) The new mean[pic]

The new standard deviation = 4

(b) The new mean[pic]

The new standard deviation = 4(1 ( 5%)

= 3.8

(c) Because originally the student’s score is equal to the mean, changing it to 0 will make the distribution of data less concentrated about the mean. As a result, the standard deviation will increase.

Quick Practice 4.17 (p. 4.58)

a) Range of Tom’s marks

[pic]

Standard deviation of Tom’s marks

= 5.54 (cor. to 3 sig. fig.)

(b) (i) Since Tom’s mark in Test 5 is lower than the lowest mark of the first 4 tests, the range of his marks will increase.

(ii) Standard deviation of Tom’s marks in the 5 tests

[pic] (cor. to 3 sig. fig.)

Hence, the standard deviation of Tom’s marks increases when compared with that in (a).

Further Practice

Further Practice (p. 4.15)

1. (a) (i) Range of the starting salaries for Group A

[pic]

Range of the starting salaries for Group B

[pic]

(ii) Since the range of the starting salaries for

Group A > the range of the starting salaries for Group B, the starting salaries for Group A are more dispersed.

(b) (i) Inter-quartile range of the starting salaries for Group A

[pic]

Inter-quartile range of the starting salaries for Group B

[pic]

(ii) ∵ IQR for Group A < IQR for Group B

∴ Based on the inter-quartile ranges of the starting salaries of the graduates, graduates of Group B have a greater dispersion in their starting salaries.

2. (a) [pic]

(b) From the graph, the median[pic]

From the graph, the range[pic]

∵ The lower quartile is the value corresponding to the cumulative frequency of 10.

∴ From the graph, the lower quartile[pic].

∵ The upper quartile is the value corresponding to the cumulative frequency of 30.

∴ From the graph, the upper quartile[pic].

Inter-quartile range[pic]

Further Practice (p. 4.23)

1. (a) Since the box for Mathematics test is the longest, the inter-quartile range of marks in Mathematics test is the largest.

(b) Since the length of the whole box-and-whisker diagram for Chinese test is the shortest, the range of marks in Chinese test is the smallest.

(c) From reading the bars inside the boxes in the diagrams, we can see that the median mark in Mathematics test is the highest.

(d) In Mathematics test, Peter’s result is below the upper quartile. In Chinese test, Peter’s result is equal to the upper quartile. In English test, Peter’s result is above the upper quartile. Therefore, Peter performs the best in English test relative to his classmates.

2. ∵ Smallest datum ’ 20

Largest datum ’ 100

Median ’ 72

The lower quartile is the value corresponding to the cumulative frequency of 12.5.

From the graph, the lower quartile ’ 60

The upper quartile is the value corresponding to the cumulative frequency of 37.5.

From the graph, the upper quartile ’ 84

∴ The required box-and-whisker diagram is:

[pic]

3. (a) From the diagram, we can see that Paul’s score is equal to the upper-quartile of Group A. Therefore, Paul’s score is higher than the scores of 75% of the students in his own group. Similarly, Mary’s score is equal to the median of Group B. Therefore, Mary’s score is higher than the scores of 50% of the students in her own group. Hence, Paul’s performance is relatively better.

(b) Let x be the number of students in each group.

Number of students having marks lower than 60 in Group A ’ 0.75x

Number of students having marks lower than 60 in Group B ’ 0.5x

∴ The required percentage

[pic]

Further Practice (p. 4.35)

1. (a) Standard deviation[pic]

| (b) |(i) |Data |1 – 10 |11 – 20 |

|Frequency |7 |9 |10 |2 |

New mean age[pic]

New standard deviation of the ages of the remaining employees[pic]

11. (a) (i) Mean mark[pic]

(ii) Standard deviation of the marks[pic]

(b) ∵ Standard deviation of school A > Standard deviation of school B

∴ School A’s result is more dispersed.

12. To obtain the largest standard deviation, the data should be dispersed from the mean as much as possible. Hence, we should choose the maximum and the minimum numbers only.

The required data set ’ {0, 0, 0, 0, 0, 10, 10, 10, 10, 10}

The largest possible value of standard deviation[pic]

13. (a) Mean waiting time[pic]

(b) Standard deviation of waiting time

[pic]

(c) No. Although the mean waiting time has been reduced to 8 min, the standard deviation is relatively large. In other words, the new distribution of waiting time is more dispersed than before and extreme values may be more likely to appear.

14. (a) ∵ The temperature of city A is always higher than that of city B.

∴ City A is warmer.

(b) Range should be used since extreme temperatures are the most important information.

(c) Range of city A ’ (32 – 27)(C = 5(C

Range of city B ’ (27 – 20)(C = 7(C

∴ City B has a greater variation in temperature on that day.

15. (a) He made a good choice. The standard deviation takes all data into account and therefore, it is the most reliable measure. Moreover, it can be applied in further statistical calculations and analysis.

(b) Standard deviation[pic]

(c) The machine does not function property since the standard deviation is greater than 0.1 kg.

Exercise 4D (p. 4.49)

Level 1

1. (a) Let x be the standard deviation of the class.

[pic]

∴ The standard deviation of the class is 6.

(b) Let y be the marks of Sam.

[pic]

∴ The marks of Sam is 65.

(c) Let z be the mean of the class.

[pic]

∴ The mean of the class is 60.

(d) Standard score of Sam[pic]

2. Mary’s standard score[pic]

3. Let x be the number of fish Jacky caught.

[pic]

∴ Jacky caught 67 fish.

4. (a) ∵ [pic]

∴ The required percentage

[pic]

(b) ∵ [pic]

∴ The required percentage

[pic]

(c) ∵ [pic]

∴ The required percentage

[pic]

(d) ∵ [pic]

∴ The required percentage

[pic]

(e) ∵ [pic]

∴ The required percentage

[pic]

5. (a) ∵ [pic]

∴ The required percentage

[pic]

The required number of females

[pic]

(b) ∵ [pic]

∴ The required percentage

[pic]

The required number of females

[pic]

6. (a) Sales of salesman A was below the mean sales of the company last month because his standard score is less than 0.

(b) Salesman A made more sales last month as his standard score is larger than salesman B’s.

7. (a) Standard score of Mary

[pic]

(b) The higher standard score a competitor obtains, the longer finishing time he/she has made. Hence, a higher standard score means a worse performance.

8. (a) David’s standard score in test 1

[pic]

David’s standard score in test 2

[pic]

(b) David performs relatively better in test 1. It is because his standard score in test 1 is higher.

Level 2

9. (a) Let x be the standard deviation of the number of tickets sold.

[pic]

∴ The standard deviation of the number of tickets sold is 5.

(b) Let y be the number of tickets Tom sold.

[pic]

∴ Tom sold 89 tickets.

10. [pic]

The percentage of boxes of washing powder weighing between 4.8 kg and 4.9 kg

[pic]

Let x be the total number of boxes of washing powder.

[pic]

∴ There are 2000 boxes of washing powder.

Percentage of boxes of washing powder weighing not less than 4.8 kg

[pic]

The required number of boxes of washing powder

[pic]

11. ∵ [pic]

∴ [pic]

∵ [pic]

∴ [pic]

(2) – (1) : [pic]

By substituting[pic]into (2), we have

[pic]

12. (a) (i) Standard score of the weight of the baby boy

[pic]

(ii) Standard score of the weight of the baby girl

[pic]

(b) The baby boy is comparatively heavier since the standard score of the weight of the baby boy is greater.

13. (a) (i) Mean[pic]

Standard deviation[pic]

(ii) Mean[pic]

Standard deviation[pic]

(b) Adam’s standard score in geography test

[pic]

Adam’s standard score in history test

[pic]

(c) Adam performs better in geography test compared to his seven close friends since his standard score in geography test is higher than that in history test.

14. (a) Standard score in jumping

[pic]

Standard score in dressage

[pic]

Standard score in eventing

[pic]

(b) (i) Tom performs the best in jumping since his standard score in jumping is the highest among all the areas.

(ii) Tom performs the worst in eventing since his standard score in eventing is the lowest among all the areas.

15. ∵ [pic]

∴ Length of the warranty period

[pic]

16. (a) ∵ [pic]

∴ The required percentage

[pic]

(b) (i) ∵ [pic]

∴ The required probability

[pic]

(ii) The required probability

[pic]

(iii) Let x be the total number of bottles of candles that should be produced.

[pic]

∴ The total number of bottles of candies is

3200.

17. (a) (i) ∵ [pic]

∴ The required probability

[pic]

(ii) ∵ [pic]

∴ The required probability

[pic]

(b) [pic]

18. (a) Connie’s standard score

[pic]

(b) Let x be Connie’s adjusted mark.

[pic]

∴ Connie’s adjusted mark is 50.4.

19. [pic]

(or any other reasonable answers)

20. (a) For machine A,

[pic]

∴ Percentage of ball bearings produced with diameters between 5.88 mm and 6.12 mm

’ 95%

For machine B,

[pic]

∴ Percentage of ball bearings produced with diameters between 5.88 mm and 6.12 mm

[pic]

∴ Machine B will produce more ball bearings with acceptable diameters.

(b) For machine A,

[pic]

∴ Percentage of ball bearings produced with diameters between 5.94 mm and 6.06 mm

’ 68%

For machine B,

[pic]

∴ Percentage of ball bearings produced with diameters between 5.94 mm and 6.06 mm

( 50%

∴ Machine A will produce more ball bearings with acceptable diameters.

(c) (i) To produce at least 95% acceptable ball bearings,

[pic]

(1) + (2) : [pic]

Substitute [pic]into (1).

[pic]

The standard deviation should be adjusted to 0.03 mm.

The minimum cost of fixing machine A

[pic]

(ii) ∵ [pic]

∴ There is no need to adjust[pic].

To produce at least 95% acceptable ball bearings,

[pic]

The mean diameter should be adjusted to

5.98 mm.

The minimum cost of fixing machine B

[pic]

Exercise 4E (p. 4.59)

Level 1

1. (a) Range : unchanged

Standard deviation : decrease

(b) Range : increase

Standard deviation : increase

(c) Range : unchanged

Standard deviation : increase

(d) Range : unchanged

Standard deviation : decrease

2. (a) Range of set B[pic]

(b) Range of set C[pic]

3. (a) (i) Mean[pic]

Standard deviation

[pic]

(b) (i) New mean[pic]

New standard deviation[pic]

(ii) New mean[pic]

New standard deviation

[pic]

4. (a) (i) Range[pic]

Standard deviation[pic]

(ii) Range[pic]

Standard deviation[pic]

(b) (i) Range = 6 ( ((5) = 11

Mean[pic]

Standard deviation

[pic]

(ii) From (a), take[pic]

Range[pic]

Standard deviation[pic]

5. The new inter-quartile range

[pic]

6. (a) Range[pic]

[pic]

(b) Standard deviation of the ages of students

[pic]

(c) New range[pic]

New IQR[pic]

New standard deviation[pic] (cor. to 3 sig. fig.)

7. (a) Range[pic]

IQR[pic]

b) Since each datum is reduced by 5 min, the range and the IQR will remain unchanged.

8. (a) Mean[pic]

Standard deviation[pic]

(b) (i) Increase

(ii) New mean[pic]

New standard deviation

[pic]

∴ The standard deviation increases.

Level 2

9. The mean of the data set ’ 19.2 (cor. to 3 sig. fig.)

∵ The range remains unchanged.

∴ The minimum and the maximum datum are not the possible values of x.

∵ The standard deviation increases.

∴ The data are more dispersed from the mean.

∴ 18 and 21 are the possible values of x.

10. (a) Class B

(b) If all marks are multiplied by 10, the ranges, the IQRs and the standard deviations will be all multiplied by 10.

11. (a) [pic]

Standard deviation

[pic]

(b) ∵ More tickets are sold at the most expensive price.

∴ The distribution of the data is more dispersed.

∴ The mean and the standard deviation increase.

12. (a) New mean[pic]

New standard deviation[pic]

(b) New mean[pic]

New standard deviation[pic]

(c) ∵ The salaries of the ten employees are equal to the mean.

∴ The distribution of the data is less concentrated.

∴ The standard deviation increases.

13. (a) Range[pic]

IQR[pic]

(b) (i) Range[pic]

IQR [pic]

(ii) Highest temperature[pic]

Lowest temperature[pic]

[pic]

[pic]

14. (a) Group B

(b) (i) For group A,

[pic]

[pic]

[pic]

inter-quartile range

[pic]

For group B,

[pic]

[pic]

[pic]

inter-quartile range

[pic]

(ii) No

(c) (i) The standard deviation will increase in group A since the leaving member has age equal to the mean. The data are relatively less concentrated about the mean.

(ii) The standard deviation will decrease in group B since the joining member has age equal to the mean. The data are relatively more concentrated about the mean.

(d) The standard deviation will increase since the new member’s age is not close to the mean. The data are relatively less concentrated about the mean.

(e) The standard deviation will decrease since the two leaving members have ages not close to the mean. The data are relatively more concentrated about the mean.

15. (a) (i) New mean[pic]

New standard deviation[pic]

(ii) The range and the IQR will be multiplied by[pic]. Hence, both decrease by 25%.

(b) (i) Sum of scores of the students

[pic]

Correct mean

[pic]

(ii) The standard deviation will increase. Since 50 is closer to the mean than 60, correcting the score will make the distribution less concentrated about the mean.

Revision Exercise 4 (p. 4.66)

Level 1

1. (a) Range[pic]

IQR[pic]

(b) Range[pic]

[pic]

(c) Range[pic]

[pic]

(d) Range[pic]

[pic]

2. (a) Mean[pic]

Standard deviation

[pic]

(b) Mean[pic]

Standard deviation

[pic]

(c) Mean

[pic]

Standard deviation

[pic]

3. (a) Range[pic]

(b) IQR[pic]

(c) Median[pic]

4. (a) Median[pic]

(b) Range[pic]

IQR[pic]

5. (a) Range[pic]

(b) Median[pic]

Lower quartile[pic]

Upper quartile[pic]

6. (a) Party B

(b) Party A

(c) Party B

7. (a) Median[pic]

(b) Range[pic]

IQR[pic]

(c)[pic]

d) The median is closer to the minimum than to the maximum. In other words, the distribution of the data in the lower half is less dispersed than that in the upper half.

8. (a) Median[pic]

Lower quartile[pic]

Upper quartile[pic]

(b)[pic]

(c) The required interval is between 49 m3 and 88 m3.

9. (a) City A has a population with more dispersed income.

(b) In city A, the monthly salary of the job offer is greater than the mean monthly income of a person by $500. Such difference is very small when compared with the standard deviation $3000. In other words, the monthly salary given to Lucy is not high.

In city B, the monthly salary of the job offer is greater than the mean monthly income of a person by $2000. Such difference is very large when compared with the standard deviation $800. In other words, the monthly salary given to Lucy is very high.

∴ Lucy should take the job offer in city B.

10. (a)

|Class mark |29.5 |49.5 |69.5 |89.5 |109.5 |

|($1000) | | | | | |

|Frequency |12 |x |y |15 |9 |

[pic]

[pic]

(b) Standard deviation[pic]

11. An engineer always needs the most accurate and reliable measure. Hence, all data must be taken into account and standard deviation should be used.

12. If the statistician wants to do accurate statistical calculations or reliable analysis, standard deviation should be used as it takes all data into account. If the statistician does not want complicated calculations, inter-quartile range is good enough for the purpose as it is not affected by extreme values.

13. (a) Range[pic]

[pic]

Inter-quartile range[pic]

Standard deviation[pic]

(b) (i) ∵ x is added to each datum.

∴ The standard deviation will remain unchanged.

∴ The standard deviation[pic]

(ii) ∵ Each datum is doubled.

∴ The standard deviation will be doubled.

∴ The standard deviation[pic]

(iii) ∵ Each datum is divided by 4.

∴ The standard deviation will be divided by 4.

∴ The standard deviation[pic]

14. (a) [pic]

[pic]

(b) ∵ [pic]

∴ [pic]

[pic]

15. (a) Mean[pic]

Standard deviation[pic] (cor. to 3 sig. fig.)

(b) The datum 2200 cc is close to the mean. So, if the manufacturer stops producing a 2200 cc model, the distribution of the data will be less concentrated about the mean. Therefore, the standard deviation will increase.

16. (a) Mean[pic]

Standard deviation[pic]

(b) Standard score[pic]

17. (a) For piano group,

mean[pic]

standard deviation

[pic]

For violin group,

mean[pic]

standard deviation

[pic]

For flute group,

mean[pic]

standard deviation

[pic]

(b) Ben, Jess and Marcy are the best participants in their own groups. Among these 3 participants, the one with the highest standard score should be the overall champion.

Standard score of Ben

[pic]

Standard score of Jess

[pic]

Standard score of Marcy

[pic]

∴ Jess will win the overall champion.

18. (a) (i) ∵ [pic]

∴ The required percentage

[pic]

(ii) ∵ [pic]

∴ The required percentage

[pic]

(b) Percentage of pigs which are either heavier than

135 kg or lighter than 90 kg

[pic]

The required number of pigs

[pic]

19. The two groups have equal mean, equal modal class and equal median. The standard deviation for group A is greater than that of group B.

20. (a) The required set of data

’ {1, 2, 3, 4, 5} or {2, 4, 6, 8}

(or any other reasonable answers)

(b) The required set of data

’ {0, 0, 0, 0, 1} or {0, 0, 0, 3, 4}

(or any other reasonable answers)

21. (a) [pic]

[pic]

(b) ∵ The data set in (b) can be formed by adding a common constant 10 to each datum of the data set in (a).

∴ The minimum, maximum, median, lower quartile and upper quartile of data set in (b) should be also increased by the same common constant 10.

∴ [pic]

[pic]

(c) (i) [pic]

(ii) [pic]

22. (a) ∵ [pic]

∴ The required probability

[pic]

(b) [pic]

The probability that the weight of a tomato is greater than or equal to[pic] which is very low. In other words, the probability of getting a tomato with weight more than 280 g is even lower.

∴ It is very unlikely that a tomato weighs more than 280 g.

Level 2

|23. |(a) |Years of experience less |Cumulative frequency |

| | |than | |

| | |0 |0 |

| | |5 |4 |

| | |10 |7 |

| | |15 |19 |

| | |20 |27 |

| | |25 |29 |

| | |30 |30 |

(b) [pic]

(c) Median[pic] (cor. to the nearest integer)

[pic], [pic]

[pic] (cor. to the nearest integer)

24. (a) For group A,

median ’ the 16th datum

’ 28 min

Q1 ’ the 8th datum

’ 10 min

Q3 ’ the 23rd datum

’ 34 min

For group B,

median = the 15th datum

= 26 min

[pic]

[pic]

The required diagrams are:

[pic]

(b) (i) Group A

(ii) Group B

iii) Group A

25. (a) Median[pic]

[pic], [pic]

[pic]

(b) (i)[pic]

(ii) No. The diagrams only suggest that in general the members read more books. Since the diagrams cannot show the actual number of books read by each member, the president cannot guarantee every member reads more books after joining the club.

26. (a) [pic]

(b) [pic]

(c) ∵ [pic]

∴ Set X is more dispersed.

27. (a) (i) For battery A,

[pic], [pic]

Inter-quartile range[pic]

For battery B,

[pic], [pic]

Inter-quartile range[pic]

(ii) Battery B

b) Yes. The middle 50% of the data of battery B are really more dispersed than that of battery A. Since the inter-quartile range focuses on the distribution of the middle 50% of given data, it is a suitable measure of dispersion in this case.

28. (a) Standard score for May

[pic]

Standard score for John

[pic]

b) The lower standard score a visitor obtains, the faster he/she reacts.

∴ John reacts relatively faster when compared with other members in his own group.

29. (a) ∵ Karen’s standard score in paper I > Karen’s standard score in paper II

∴ Karen performs better in paper I compared with the scores of the whole class.

(b) [pic]

[pic]

c) Karen’s score in the overall Mathematics examination

[pic]

30. Let[pic]and ( be the new mean and the new standard deviation of the test results respectively.

For Bob,[pic] ......(1)

For Kate,[pic] ......(2)

(1) – (2), we have

[pic]

By substituting ( ’ 8 into (2), we have

[pic]

∴ The new mean and the new standard deviation of the test results are 60 and 8 respectively.

31. (a) Mean price

’$4840 (cor. to 3 sig. fig.)

Standard deviation

’ $4360 (cor. to 3 sig. fig.)

The price should be between

$(4840 – 4360) and $(4840 + 4360),

i.e. $480 and $9200.

∴ Mr. Mok can choose watches B, C, D, E and F.

(b) The required percentage

[pic]

c) If the prices are normally distributed, there should be 68% of the data lying within one standard deviation from the mean. However, there are only 55.6% of the watches with prices lying within one standard deviation from the mean. Therefore, the prices of the watches do not follow the normal distribution.

32. (a) ∵ 69.5 mm ’ [70 – 2(0.25)] mm ’[pic]

70.5 mm ’ [70 + 2(0.25)] mm ’[pic]

∴ The required percentage

[pic]

b) Net profit

[pic]

33. (a) (i) ∵ 1.3 kg ’ [1 + 2(0.15)] kg ’[pic]

∴ The percentage of chickens which weigh over 1.3 kg

[pic]

(ii) ∵ 1.3 kg ’ (1.2 + 0.1) kg ’[pic]

∴ The percentage of ducks which weigh over

1.3 kg

[pic]

b) P(at least one weighs over 1.3 kg)

[pic]

34. (a) ∵ 1595 lm ’ [1700 – 3(35)] lm ’[pic]

1770 lm ’ [1700 + 2(35)] lm =[pic]

∴ Percentage of light bulbs having brightness between 1595 lm and 1770 lm

[pic]

∴ The required number of light bulbs

[pic]

b) Let y be the number of light bulbs which should be manufactured in total.

[pic]

∴ 10 000 light bulbs should be manufactured in total.

(c) ∵ 1595 lm ’ [1700 – 3(35)] lm ’[pic]

∴ Percentage of light bulbs having brightness lower than 1595 lm

[pic]

P(more than one light bulb has brightness lower than 1595 lm)

’ P(2 light bulbs have brightness lower than 1595 lm) + P(all light bulbs have brightness lower than 1595 lm)

[pic]

35. (a) (i) If the two records are corrected, the sum of heights of the 30 students will be decreased by 20 cm.

∴ The mean will be decreased by [pic]

ii) Median[pic]

If both incorrect data are 171 cm or 173 cm, then after the corrections, they become 161 cm or 163 cm, which are still not the 15th and the 16th datum. So, the median remains unchanged.

(iii) The original mode is 157 cm, with a frequency of 5. If both incorrect data are 171 cm or 173 cm, then after the corrections, they become 161 cm or 163 cm, with a frequency less than 5. Therefore, the mode remains unchanged.

(iv) If both incorrect data are 173 cm, then after the corrections, the largest datum is 171 cm which is less than the original one by 2 cm, and hence, the range will be decreased by 2 cm. Otherwise, the range remains unchanged.

(v) Before the corrections,

Q1 ’ 155 cm, Q3 ’ 167 cm

After the corrections,

[pic]

Since[pic]the IQR will be decreased by 2 cm.

b) Originally, the distribution of the data is concentrated about the interval 155 cm ( 167 cm, i.e. the middle 50% of the data.

No matter what corrections will be done, the two incorrect data will certainly fall in the interval

155 cm ( 165 cm. Precisely, from (a)(v), the IQR will be decreased by 2 cm. So, the new distribution is more concentrated than before.

In other words, the data are less dispersed.

36. (a) (i) Range[pic]

(ii) [pic]

Inter-quartile range[pic]

(i) Standard deviation[pic]

(b) The range will be increased to 4 since the highest age is increased by 1.

The inter-quartile range will remain unchanged since the 37th datum will be the same as before.

The standard deviation will be increased since the new datum is not close to the mean and relatively less data are concentrated about the mean.

37. (a) Mean[pic]

Median[pic]

Inter-quartile range = the 23rd datum ( the 8th datum

= 3 ( 1

=[pic]

Standard deviation[pic]

b) The inter-quartile range will remain unchanged since the new and original lower and upper quartiles are the same. The standard deviation will increase since the removed data are all equal to the mean and relatively less data are concentrated about the mean.

(c) The inter-quartile range will remain unchanged since the new and original lower and upper quartiles are the same. The standard deviation will increase since the new datum is not close to the mean and relatively less data are concentrated about the mean.

38. (a) ∵ [pic] and [pic]

∴ [pic]

(b) ∵ [pic]

∴ Standard deviation

[pic]

39. ∵ Mean[pic]

and standard deviation = 3

∴ [pic]

[pic] [pic]

Multiple Choice Questions (p. 4.76)

1. Answer: A

Arrange the heights (cm) in ascending order:

[pic]

∴ Inter-quartile range of the heights

[pic]

2. Answer: A

From the graph,

[pic]

∴ Inter-quartile range[pic]

3. Answer: B

Mean[pic]

Standard deviation

[pic]

4. Answer: B

∵ Each datum is increased by 10.

∴ Median is increased by 10 while range and inter-quartile range will remain unchanged.

∴ Only I and II are correct.

5. Answer: C

The data set[pic]can be formed by adding the common constant a to each datum of data set[pic]

The data set[pic]can be formed by multiplying each datum of[pic]by the common constant d.

Obviously, the mean and the standard deviation of [pic]are 0 and[pic]respectively. So, the required standard deviation is[pic]

6. Answer: A

From the diagram,

median of data set A ’ median of data set B

range of data set A > range of data set B

inter-quartile range of data set A < inter-quartile range of

data set B

∴ Only I and II are true.

7. Answer: B

From the diagram,

∵ The length of the box for Chinese examination is the shortest.

∴ Chinese examination has the smallest inter-quartile range of marks.

8. Answer: D

∵ Each datum is decreased by 5%.

∴ The mean and the standard deviation of the set of data will also be decreased by 5%.

∴ The new mean[pic]

The new standard deviation[pic]

9. Answer: B

Since all the salaries of the leaving employees are equal to the mean, removing them will not change the mean.

Moreover, the distribution of data will be less concentrated about the mean. As a result, the standard deviation will increase.

10. Answer: B

From the graph,

range of P ’ range of Q

inter-quartile range of P < inter-quartile range of Q

median of P ’ median of Q

∴ Only II is true.

11. Answer: C

For I, standard deviation ’ 14.1 (cor. to 3 sig. fig.)

For II, standard deviation ’ 11.5 (cor. to 3 sig. fig.)

For III, standard deviation ’ 16.0 (cor. to 3 sig. fig.)

∴ II < I < III

12. Answer: A

Let x marks be the standard deviation.

[pic]

∴ The standard deviation is 5 marks.

13. Answer: C

∵ [pic]

∴ The required percentage

[pic]

Investigation Corner (p. 4.81)

1. Consider the following table.

|Marks (x) |Class Mark |Number of |Number of |

| | |candidates in |candidates in|

| | |2008 |2009 |

|[pic] |[pic] |1260 |370 |

|[pic] |[pic] |6745 |3100 |

|[pic] |[pic] |16 970 |21 075 |

|[pic] |[pic] |17 035 |21 890 |

|[pic] |[pic] |6710 |3205 |

|[pic] |[pic] |1280 |360 |

Mean in 2008

[pic]

Standard deviation in 2008

[pic]

Mean in 2009

[pic]

Standard deviation in 2009

[pic]

2. For 2008,

passing mark[pic]

Lowest mark for a distinction

[pic]

For 2009,

passing mark[pic]

Lowest mark for a distinction

[pic]

3. Under system Y in 2008,

lowest mark for ‘B’

[pic]

lowest mark for ‘D’

[pic]

|Candidate |Year |Marks |Grade |

| | | |System X |System Y |

|Anthony |2009 |87 |B |A |

|Rachel |2008 |36 |F |E |

|Tim |2009 |42 |E |F |

4. (a) Under system Y, since the mean[pic]in 2008 and 2009 are very close but the standard deviation[pic]in 2009 is much less than that in 2008, the mark for a distinction in 2009 is much lower than that in 2008.

(b) By using the same argument in (a), the passing mark in 2009 is much higher than that in 2008.

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