Chapter 17 Uses And Abuses Of Statistics
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17 Uses and Abuses of Statistics
17 Uses and Abuses of Statistics • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Activity Activity 17.1 (p. 106)
reasonable answers) (d) Average income of a household in a city (or any other reasonable answers)
1.
p. 123
(a) Census and Statistics Department (b) To investigate the unemployment rate in January – March, 20XX.
2.
(a) 24 000 households (b) No, because it is difficult to cover the total labour force.
3.
(a) Interview (b) Detailed information can be obtained.
1.
Sampling method is misused. Simple random sampling should be used instead of systematic sampling.
2.
No misuse of methods.
3.
Data collection method is misused. Questionnaire should be used instead of interview.
4.
Data collection method is misused. Interview on randomly selected people waiting at the stops should be used instead.
Activity 17.2 (p. 121) 1.
2.
3.
4.
Possible errors: Low validity and errors may be induced. Improvements: Recruit more observers, possibly use video to record the events for further observations. Possible errors: Interview is not a suitable method to collect sensitive data. Only conservative answers might be obtained. Improvements: Anonymous questionnaires should be used. Possible errors: Low response rate may be resulted because questionnaires may be sent to wrong targets. People with strong opinions about the quality are more likely to return the questionnaire and may bias the results. Improvements: Questionnaires should be sent to the users only.
Exercise Exercise 17A (p. 123) Level 1 1. (a) (b) (c) (d) (e) (f) 2.
(a) (b) (c) (d) (e) (f) (g) (h)
interview / questionnaire experiment questionnaire observation statistical publication / the Internet interview / questionnaire experiment statistical publication / the Internet
3.
(a)
P (select a book) =
Possible errors: Low validity of results as experiments are carried out on different appliances. Improvements: Experiments should be carried out on the same appliance.
Follow-up Exercise
systematic sampling stratified random sampling simple random sampling non-probability sampling
p. 118 (a) Average number of people waiting at a bus stop (or any other reasonable answers) (b) Customer’s opinion of a shopping mall (or any other reasonable answers) (c) Favourite food of a class of student (or any other
25 4 1 × = 100 100 100
(b) simple random sampling
p. 115 (a) (b) (c) (d)
systematic sampling stratified random sampling simple random sampling systematic sampling stratified random sampling non-probability sampling
4.
(a) All the vehicles using the highway in a day. (b) observation
Level 2 5. (a) Simple random sampling. Assign a number to each student in the school (or use their unique student numbers) and then randomly select 300 students for the survey. 300 3 (b) P (select a student) = 1000 = 10
Certificate Mathematics in Action Full Solutions 5B
(c) interview / questionnaire 6.
7.
(a) We can use age, sex, occupation, marital status or educational background as a stratifying variable, since all these factors may affect the income of an individual. An individual can only process one property in all these stratifying variables, e.g. male or female, single or married, etc. (b) interview / questionnaire (a) With the presence of the observer, the group behaviour can change and give a biased result. The observer should observe passively. (b) The sampling method is biased because members of the population do not have the same chance of being selected. Simple random sampling on the whole population should be used. (c) The sampling method is misused because car-owners and non car-owners may have different opinions. Stratified random sampling should be used to give a better result.
Revision Exercise 17 (p. 131) Level 1 1. (a) (b) (c) (d) 2.
3.
(a) (b) (c) (d) (e) (f)
simple random sampling systematic sampling stratified random sampling stratified random sampling interview / questionnaire interview / questionnaire experiment questionnaire statistical publication / the Internet statistical publication / the Internet
those did not like Mr. Lo, were more likely to return their ballots. 6.
First of all, the data are presented poorly. They are divided into too many groups, which are both time-consuming for data tabulation and difficult in 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, have school bags heavier than 6.2 kg. The conclusion should point out the weights of the school bags lie between 5.0 kg and 7.3 kg, with most of them lie between 5.6 kg and 6.7 kg.
7.
The statement is not quite valid. The car manufacturer regards the customers who did not give a negative response as ‘satisfied’ customers. This would mislead and give a wrong impression to the customers.
Multiple Choice Questions (p. 133) 1.
Answer: C
2.
Answer: B
3.
Answer: A
4.
Answer: C 6 6+4 6 = 50 × 10 = 30
Number of boys = 50 ×
5.
Interview is not suitable for collecting sensitive data. Observation is not suitable to collect qualitative data.
(a) simple random sampling (b) (i) stratified random sampling 375 × 200 = 75 375 + 625 625 Number of female staff = × 200 = 125 375 + 625 125 1 (iii) P (female staff) = 625 = 5
∴
(ii) Number of male staff =
Level 2 4. The income of staff at different levels are likely to be different. The sample selected by simple random sampling may not be proportional to the population and thus biases the result. In this case, stratified random sampling with staff levels as the stratifying variable should be used. 5.
Although the sample has covered 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
Answer: B
Only II is suitable.
6.
Answer: A For I and II, all the customers would have equal chance to be selected while III would not. ∴ I and II would give a good representative sample.
7.
Answer: D For I and II, the stratifying variable can stratify the students into only one stratum. For III, since a student can have more than one club and society membership, therefore it cannot stratify the students into only one stratum. ∴ Only I and II are appropriate stratifying variables.
8.
Answer: D For I, II and III, they are all possible reasons.
17 Uses and Abuses of Statistics • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Activity Activity 17.1 (p. 106)
reasonable answers) (d) Average income of a household in a city (or any other reasonable answers)
1.
p. 123
(a) Census and Statistics Department (b) To investigate the unemployment rate in January – March, 20XX.
2.
(a) 24 000 households (b) No, because it is difficult to cover the total labour force.
3.
(a) Interview (b) Detailed information can be obtained.
1.
Sampling method is misused. Simple random sampling should be used instead of systematic sampling.
2.
No misuse of methods.
3.
Data collection method is misused. Questionnaire should be used instead of interview.
4.
Data collection method is misused. Interview on randomly selected people waiting at the stops should be used instead.
Activity 17.2 (p. 121) 1.
2.
3.
4.
Possible errors: Low validity and errors may be induced. Improvements: Recruit more observers, possibly use video to record the events for further observations. Possible errors: Interview is not a suitable method to collect sensitive data. Only conservative answers might be obtained. Improvements: Anonymous questionnaires should be used. Possible errors: Low response rate may be resulted because questionnaires may be sent to wrong targets. People with strong opinions about the quality are more likely to return the questionnaire and may bias the results. Improvements: Questionnaires should be sent to the users only.
Exercise Exercise 17A (p. 123) Level 1 1. (a) (b) (c) (d) (e) (f) 2.
(a) (b) (c) (d) (e) (f) (g) (h)
interview / questionnaire experiment questionnaire observation statistical publication / the Internet interview / questionnaire experiment statistical publication / the Internet
3.
(a)
P (select a book) =
Possible errors: Low validity of results as experiments are carried out on different appliances. Improvements: Experiments should be carried out on the same appliance.
Follow-up Exercise
systematic sampling stratified random sampling simple random sampling non-probability sampling
p. 118 (a) Average number of people waiting at a bus stop (or any other reasonable answers) (b) Customer’s opinion of a shopping mall (or any other reasonable answers) (c) Favourite food of a class of student (or any other
25 4 1 × = 100 100 100
(b) simple random sampling
p. 115 (a) (b) (c) (d)
systematic sampling stratified random sampling simple random sampling systematic sampling stratified random sampling non-probability sampling
4.
(a) All the vehicles using the highway in a day. (b) observation
Level 2 5. (a) Simple random sampling. Assign a number to each student in the school (or use their unique student numbers) and then randomly select 300 students for the survey. 300 3 (b) P (select a student) = 1000 = 10
Certificate Mathematics in Action Full Solutions 5B
(c) interview / questionnaire 6.
7.
(a) We can use age, sex, occupation, marital status or educational background as a stratifying variable, since all these factors may affect the income of an individual. An individual can only process one property in all these stratifying variables, e.g. male or female, single or married, etc. (b) interview / questionnaire (a) With the presence of the observer, the group behaviour can change and give a biased result. The observer should observe passively. (b) The sampling method is biased because members of the population do not have the same chance of being selected. Simple random sampling on the whole population should be used. (c) The sampling method is misused because car-owners and non car-owners may have different opinions. Stratified random sampling should be used to give a better result.
Revision Exercise 17 (p. 131) Level 1 1. (a) (b) (c) (d) 2.
3.
(a) (b) (c) (d) (e) (f)
simple random sampling systematic sampling stratified random sampling stratified random sampling interview / questionnaire interview / questionnaire experiment questionnaire statistical publication / the Internet statistical publication / the Internet
those did not like Mr. Lo, were more likely to return their ballots. 6.
First of all, the data are presented poorly. They are divided into too many groups, which are both time-consuming for data tabulation and difficult in 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, have school bags heavier than 6.2 kg. The conclusion should point out the weights of the school bags lie between 5.0 kg and 7.3 kg, with most of them lie between 5.6 kg and 6.7 kg.
7.
The statement is not quite valid. The car manufacturer regards the customers who did not give a negative response as ‘satisfied’ customers. This would mislead and give a wrong impression to the customers.
Multiple Choice Questions (p. 133) 1.
Answer: C
2.
Answer: B
3.
Answer: A
4.
Answer: C 6 6+4 6 = 50 × 10 = 30
Number of boys = 50 ×
5.
Interview is not suitable for collecting sensitive data. Observation is not suitable to collect qualitative data.
(a) simple random sampling (b) (i) stratified random sampling 375 × 200 = 75 375 + 625 625 Number of female staff = × 200 = 125 375 + 625 125 1 (iii) P (female staff) = 625 = 5
∴
(ii) Number of male staff =
Level 2 4. The income of staff at different levels are likely to be different. The sample selected by simple random sampling may not be proportional to the population and thus biases the result. In this case, stratified random sampling with staff levels as the stratifying variable should be used. 5.
Although the sample has covered 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
Answer: B
Only II is suitable.
6.
Answer: A For I and II, all the customers would have equal chance to be selected while III would not. ∴ I and II would give a good representative sample.
7.
Answer: D For I and II, the stratifying variable can stratify the students into only one stratum. For III, since a student can have more than one club and society membership, therefore it cannot stratify the students into only one stratum. ∴ Only I and II are appropriate stratifying variables.
8.
Answer: D For I, II and III, they are all possible reasons.
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