Chapter 7 Full Solutions



5 Uses and Abuses of Statistics

Review Exercise 5 (p. 5.4)

1. (a) observation

(b) experiment

(c) questionnaire / interview

(d) interview

2. (a) Advantage: low cost

Disadvantage: misunderstanding of questions

(or any other reasonable answers)

(b) Advantage: high response rate

Disadvantage: high cost

(or any other reasonable answers)

(c) Advantage: immediate responses

Disadvantage: lower response rate than street

interview

(or any other reasonable answers)

3. Yes. The vertical axis of this graph does not start from zero. It misleads readers into thinking that the profit of the company has grown significantly in the past 5 years.

4. (a) Ratio of the monthly sales of brand A to brand B

= 400 000 : 200 000

=[pic]

(b) Ratio of the areas of packages drawn for brand A to brand B

[pic]

(c) Yes. A bar chart can be used instead.

5. No. The original numbers of viewers are not given. The claim is not valid if the original number of viewers of Ada’s blog is much lower than that of Eddie’s. For example, the number of viewers of Ada’s blog may change from 100 to 180 while that of Eddie’s may change from 1000 to 1400.

6. No. Since brand D is a new brand of shampoo and its price is not involved in calculating the average price in 2007, the items involved in calculating the two averages are different.

Activity

Activity 5.1 (p. 5.7)

1. (a) The objective of the survey is to find out which kind of performance the students like most.

(b) interview

2. The students they interviewed are from different groups with quite different interests.

3. Yes

4. Select 100 students randomly in the morning assembly.

(or any other reasonable answers)

Activity 5.2 (p. 5.18)

|Question |Flaw(s) |

|1 |It may be difficult to recall the exact number of emails|

| |we sent in the previous month. |

|2 |1. The construction of the question (a double negative|

| |question) is difficult to understand. |

| |2. Questions starting with “Don’t you …” assume that |

| |the respondents should agree with the statements that |

| |follow. |

|3 |Some possible answers are: quite often, not very often, |

| |occasionally. They have different meanings to different |

| |people and the results are difficult to analyse. |

|4 |This is a leading question. More respondents may answer |

| |‘air-conditioning’ than if the item had not been given |

| |in the question. |

|5 |The question should avoid technical terms. Most people |

| |do not know what WYSIWYG is. (What You See Is What You |

| |Get) |

|6 |The question contains two parts: ‘convenient’ and |

| |‘economical’. Respondents cannot answer ‘yes’ to one |

| |part and ‘no’ to the other part. |

Activity 5.3 (p. 5.24)

Case 1

Limitations or problems

(a) 1. Some students may join more than one club.

2. The students who are not members of any clubs have no chance to be selected.

(b) 1. The students in the same class may take part in some activities together and hence develop similar interests.

2. The favourite activities may be different for different forms. In this method, the students of some of the forms may not be selected at all.

Case 2

Limitations or problems

(a) 1. Only the people who listen to the programme know about the survey.

2. People with strong opinions about the issue are more likely to phone in.

(b) 1. People who do not pass by an MTR station will have no chance to be selected.

2. People with friendly faces have a higher chance to be selected.

Activity 5.4 (p. 5.27)

Case 1 Problem:

It is difficult to count accurately with just one observer.

Improvement:

Recruit more observers. If possible, video-record the events for further observations.

Case 2 Problem:

There may be a low response rate. Moreover, some questionnaires may sent to non-users. Users with strong opinions about the quality are more likely to submit the questionnaires and may bias the results.

Improvement:

Questionnaires should only be sent to users of the after-sales service.

Activity 5.5 (p. 5.29)

( The performance in the beginning might have been very poor.

( The vertical scale and the exact time of ‘Past’ are missing.

( No. It should be an increase of 8% (or 4 percentage points).

( No

( No. It may be due to a comparatively larger number of students taking that subject.

( Percentage increase[pic]

Classwork

Classwork (p. 5.8)

1. sample

2. sample survey

3. non-probability sampling

4. population

5. census

Classwork (p. 5.11)

|Convenience sampling |Method 3 | |

|Simple random sampling |Method 1 |Probability of being |

| | |selected =[pic] |

|Systematic sampling |Method 4 |Interval = 5, starting |

| | |random number = 4 |

|Stratified random sampling |Method 2 |The stratifying variable |

| | |is business nature. |

Classwork (p. 5.17)

(a) average number of people waiting at a bus stop

(b) customers’ opinions of a shopping mall

(c) favourite food of a class of students

(d) effectiveness of a new light bulb

(e) average income of a household in a city

(or any other reasonable answers)

Quick Practice

Quick Practice 5.1 (p. 5.13)

(a) Obtain the name list of all the students in the school. Randomly draw the required number of names from the list.

(b) Number of boys to be interviewed[pic]

Number of girls to be interviewed[pic]

Quick Practice 5.2 (p. 5.14)

(a) Randomly select an integer n between 1 and 8. Select the 40 candidates with seat numbers n, n + 8, n + 16, … ,

n + 312.

(b) There may be some repeated patterns in conditions in the seating arrangement. For example, starting from n, every 8 seats thereafter are next to a window facing a noisy street.

Quick Practice 5.3 (p. 5.25)

1. Not every customer is a VIP card holder.

2. The VIP card holders chosen may not have visited the store after the renovation.

3. The response rate of a telephone survey is likely to be lower than that of face-to-face interviews.

(or any other reasonable answers)

Quick Practice 5.4 (p. 5.26)

(a) No. The service and food quality may be different during peak hours.

(b) Billy should choose the time and day of visits randomly.

Quick Practice 5.5 (p. 5.28)

(a) It is not appropriate to compare results of experiments carried out on different appliances.

(b) Experiments should be carried out on the same appliance.

Further Practice

Further Practice (p. 5.14)

1. Since the pollutants emitted by vehicles using different fuels vary greatly, we can stratify the vehicles according to their kinds of fuel used, like diesel, petrol, unleaded petrol, etc. Then a sample is selected proportionally according to the size of each stratum by simple random sampling. Age of vehicles or types of vehicles are also possible stratifying variables.

2. (a) No. Since the number of students in each class may not be the same, students in some classes may have a higher chance of being selected.

(b) No, this is not simple random sampling.

Exercise

Exercise 5A (p. 5.20)

Level 1

1. (a) The population of the survey is all the students in the class.

(b) The population of the survey is all the students who have had a part-time job.

(c) The population of the survey is all the employees who need to use a computer in the office.

2. (a) sample survey

(b) census

(c) census

3. (a) stratified random sampling

(b) convenience sampling

(c) systematic sampling

(d) convenience sampling

4. (a) interview / questionnaire

(b) experiment

(c) questionnaire

(d) observation

(e) interview / questionnaire

(f) experiment

(g) statistical publication / the Internet

5. (a) Q2

(b) Q1

(c) Q2

6. (a) The required probability[pic]

(b) Simple random sampling is used.

7. (a) The population of the survey is the traffic flow counts of the highway in the period under consideration.

(b) observation

Level 2

8. (a) The population of the survey is all the staff in the company.

(b) Draw out the required number of staff numbers at random.

9. (a) We can use age, sex, occupation, marital status or education background as a stratifying variable, since all these factors may affect the income of an individual. An individual can possess only one property in any of these stratifying variables, e.g. male or female, single or married, etc.

(b) interview / questionnaire

10. (a) Method 1: simple random sampling

Method 2: stratified random sampling

(b) Number of Chinese pop music CDs in the sample

[pic]

(c) Method 2. It ensures that CDs from each category are included in the sample.

11. (a) (i) Select [pic] of the students randomly from each form. The number of students selected is in proportion to the total number of students in each form.

(ii) Select a random integer between 1 and 4, say n, and select the students with student numbers n,

n + 4, n + 8, … , n + 996.

(b) Stratified random sampling:

Select [pic] of the students randomly from each form. The number of students selected is in proportion to the total number of students in each form.

Systematic sampling:

Select a random integer between 1 and 8, say n, and select the students with student numbers n, n + 8,

n + 16, … , n + 992.

(c) Stratified random sampling:

more likely to give a representative sample / ensure students from each form are included in the sample

Systematic sampling:

only one random number is required / less time consuming and less expensive

12. (a) There are two parts in this question. The respondents may be confused if they want to answer ‘yes’ to one part but ‘no’ to the other. It is also difficult to analyse because we cannot be sure for which part of the question the answers are.

It can be rewritten into two questions as follows:

Do you like wind-surfing?

Yes □ No □

Are you good at wind-surfing?

Yes □ No □

(b) The question is loaded. It assumes that the respondent supports opening more restaurants in Disneyland.

It can be modified as follows:

What kind of facilities do you think should be built to provide better services in Disneyland?

restaurants □ hotels □

barbecue sites □ swimming pools □

other (please specify: _______________ )

13. They are leading questions. A higher percentage of people will prefer small class teaching if Q1 is asked than if Q2 is asked.

14. Q1 Do you think there are enough choices of food?

Yes □ No □

Q2 What is your favourite cooked food?

chicken wing □ fish ball □

instant noodles □ sandwiches □

other (please specify: ______________ )

Q3 What do you think about the price?

very cheap □ quite cheap □

reasonable □ quite expensive □

very expensive □

Q4 Use a scale of 1 – 5 to rate the cleanliness of the tuck shop environment. (‘5’ means very clean.) ________

(or any other reasonable answers)

Revision Exercise 5 (p. 5.33)

Level 1

1. (a) probability sampling

(b) non-probability sampling

(c) probability sampling

(d) probability sampling

2. (a) simple random sampling

(b) systematic sampling

(c) stratified random sampling

(d) stratified random sampling

3. (a) interview / questionnaire

(b) interview / questionnaire

(c) experiment

(d) interview

(e) statistical publication / the Internet

(f) statistical publication / the Internet

4. (a) Q2

(b) How much on average do you spend per visit?

Under $100 □ $100 – $149.9 □

$150 – $200 □ More than $200 □

(c) People who are not Chinese may be offended by the question.

5. (a) simple random sampling

(b) (i) stratified random sampling

(ii) [pic]male staff and [pic]female staff should be selected.

(iii) P(a female staff being selected)[pic]

6. The sample is not a good representation. Some students may join more than one sports club. Also, the students who are not members of any sports clubs have no chance to be selected.

7. (a) questionnaire

(b) Before the books are sold, enclose a questionnaire in randomly selected books.

8. (a) income, sex, age (or any other reasonable answers)

(b) interview / questionnaire

Level 2

9. (a) With the presence of the observer, the group behaviour can change and give a biased result. The observer should observe without letting students know.

(b) The sampling method is biased because the characteristics of people in Central are likely to be different from those in other districts. Samples should be taken in other districts also.

10. Plan B. In plan A, the probability of being selected depends on the length of time the car is parked.

11. (a) Method 1: simple random sampling

Method 2: stratified random sampling

(b) Method 2. It ensures that staff at each level are included in the sample.

(c) more time consuming and expensive / more complicated analysis

12. (a) Maximum number of households selected

[pic]

(b) (i) Choose a random integer n between 1 and 8. Select the nth flat on each floor of the 8 buildings.

(ii) Randomly select a flat on each floor of the 8 buildings.

(c) Stratified random sampling. There may be some repeated patterns of the flats on different floors.

13. (a) Mean number of viewers per day

[pic]

(b) Actual mean number of viewers per day

[pic]

(c) No, John’s estimate is not reasonable. He can assign a weight to the data selected, i.e. [pic] for data from Monday to Thursday and [pic] for data from Friday to Sunday. Then use the weighted mean ([pic]) for the estimation. (or any other reasonable answers)

14. Although the sample contains 100 000 people of the population, the readers of the News Digest are mainly middle class. As a result, the sample is biased. Also, people having a strong opinion about the election, such as those who did not like Mr Lo, were more likely to return their ballots.

15. First of all, the data is presented poorly. It is divided into too many groups, which is time-consuming for data tabulation and difficult for studying distribution. A frequency polygon with fewer groups should be used to present the data. Secondly, the conclusion is not very significant. Only 16 students, just more than half, having school bags heavier than 6.2 kg. The conclusion should point out that the weights of school bags lie between

5.0 kg and 7.3 kg, with most of them lying between 5.6 kg and 6.7 kg.

16. The statement is not quite valid. The car manufacturer regards the customers who did not give a negative response as ‘satisfied’ customers. This will mislead potential buyers.

17. (1) The sample includes only old people who are members of elderly health centres.

(2) Their health conditions are not assessed by medical professionals.

(3) Some old people may not be able to read and therefore cannot complete the questionnaires on their own.

(4) Nothing is said about people who play mah-jong less than 3 hours per day.

(5) Old people who are able to play mah-jong regularly may have better health to begin with. Whether their good health is a result of playing mah-jong is not known.

(or any other reasonable answers)

Multiple Choice Questions (p. 5.37)

1. Answer: B

Sample size[pic]

Population size[pic]

2. Answer: C

3. Answer: A

4. Answer: C

Number of boys[pic]

5. Answer: B

Interview is not suitable for collecting sensitive data.

Observation is not suitable to collect qualitative data.

∴ Only II is suitable.

6. Answer: A

Method I and II ensure that the sample is selected across the population while method III cannot.

∴ I and II would give a good representative sample.

7. Answer: D

For I and II, the stratifying variables can stratify each student into only one stratum.

For III, since a student can have more than one club and society membership, it cannot stratify each student into only one stratum.

∴ Only I and II are appropriate stratifying variables.

8. Answer: D

9. Answer: D

Investigation Corner (p. 5.39)

Problems of the data analysis and the conclusions:

(1) It is not stated whether the relationship scale indicates mean, mode or median scores.

(2) Since the numbers of students in different sub-groups are different, the mean of the relationship scale cannot be calculated by simply taking the mean of the data 7, 6, 6 and 7.

(3) Most female students in junior forms will talk to their mothers first when they have trouble. But it does not mean that they are the only people who do so.

(or any other reasonable answers)

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